Factor Analysis, Assumptions. Exploratory Factor Analysis and Confirmatory Factor Analysis and Assumption.

Factor Analysis, Assumptions. Exploratory Factor Analysis and Confirmatory Factor Analysis and Assumption. A few Conditions and Formulae

1. Dr. A. Michael J Leo
Assistant Professor of Education
St. Xavier’s College of Education(Autonomous)
Palayamkottai – 627002
Tirunelveli District, Tamil Nadu, India
[email protected], 9994006762


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Dr. A. MIchael J Leo, Assistant Professor, St.
Xavier's College of Education
(Autonomouis), Palayamkottai - 627002

2. Factor Analysis
•A method of
correlation
in order to find and
describe the
underlying
factors for a large
set of variables

•Data Reduction
Technique
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Dr. A. MIchael J Leo, Assistant Professor, St.
Xavier's College of Education
(Autonomouis), Palayamkottai - 627002

3. Categories
•Exploratory Factor Analysis (EFA)


•Confirmatory Factor Analysis (CFA)
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Dr. A. MIchael J Leo, Assistant Professor, St.
Xavier's College of Education
(Autonomouis), Palayamkottai - 627002

4. Exploratory Factor Analysis (EFA)

•The number of constructs and the underlying
factor structure are identified.
•Hypothesizes an underlying construct, a variable
not measured directly
•Allows you to describe and identify the number of
latent constructs (factors)

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Dr. A. MIchael J Leo, Assistant Professor, St.
Xavier's College of Education
(Autonomouis), Palayamkottai - 627002

5. •To explore the possible underlying factor
structure of a set of measured variables
without imposing any preconceived structure
on the outcome (Child, 1990).
•To construct a questionnaire to measure an
underlying variable
•To solve the problem of Multicollinearity by
combining variables that are collinear

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Dr. A. MIchael J Leo, Assistant Professor, St.
Xavier's College of Education
(Autonomouis), Palayamkottai - 627002

6. EFA - Assumptions
• Interval or ratio level of measurement
• Sample needs to be independent Observations
• Relationship between observed variables is linear
• A normal distribution (each observed variable)
• Sample Should not have Outliers

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Dr. A. MIchael J Leo, Assistant Professor, St.
Xavier's College of Education
(Autonomouis), Palayamkottai - 627002

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Dr. A. MIchael J Leo, Assistant Professor, St.
Xavier's College of Education
(Autonomouis), Palayamkottai - 627002

8. Step 1(a) : Correlation Matrix
•If there is sufficient correlation among the
variables/items, the common factors could be
extracted.
•Correlation Co-efficient >=0.3 is Acceptable
Range.
•If the correlation is nil/too high b/w any two
variables, remove it.

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Dr. A. MIchael J Leo, Assistant Professor, St.
Xavier's College of Education
(Autonomouis), Palayamkottai - 627002

9. Step 1(b) : Bartlett Test of Sphericity
•Check whether the Correlation Matrix is identity
matrix or not.
•H0 : There is no significant relationship among
the variables .
•H0: It Needs to be rejected
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Dr. A. MIchael J Leo, Assistant Professor, St.
Xavier's College of Education
(Autonomouis), Palayamkottai - 627002

10. Step 1(c) : KMO test (Kaiser-Meyer-Olkin)
Sofroniou (1999)
•Marvelous: values in the .90s
•Meritorious: values in the .80s
•Middling: values in the .70s
•Mediocre: values in the .60s
•Miserable: values in the .50s
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Dr. A. MIchael J Leo, Assistant Professor, St.
Xavier's College of Education
(Autonomouis), Palayamkottai - 627002

11. Step 2: Factor Extraction
•Eigen Value : ( Should be >1)
If the number of factors are not sufficient or
Convinced, the No. of Factors could be fixed by
the formula (Fruchter, Benjamin, Op. Cit. pp. 68-69.)

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Dr. A. MIchael J Leo, Assistant Professor, St.
Xavier's College of Education
(Autonomouis), Palayamkottai - 627002

12. Communalities
•Proportions of Variance Accounted for By
Selected Components
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Dr. A. MIchael J Leo, Assistant Professor, St.
Xavier's College of Education
(Autonomouis), Palayamkottai - 627002

13. Step 2 (a): Extraction Methods
•Principal Component Analysis(PCA)
•Considers Entire variance of the Data
•Min. No. of Factors explain Max. Variance
•Principal Axis Factoring (PAF)
•Considers Common Variance of the Data
•Identifies Common Variance as factors
•PCA or PAF
•Variables/Items >=30 and communalities
>=0.6, the results will be one and the same.


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Dr. A. MIchael J Leo, Assistant Professor, St.
Xavier's College of Education
(Autonomouis), Palayamkottai - 627002

14. Step 3: Rotation
•Orthogonal Rotation :
•The resulting factors are not correlated
•Efficient but not Natural
•SPSS: Quartimax, Varimax & Equimax
•Oblique Rotation:
•Resulting factors are Correlated.
•Not efficient but Natural
1)SPSS: Direct Oblimin & Promax

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Dr. A. MIchael J Leo, Assistant Professor, St.
Xavier's College of Education
(Autonomouis), Palayamkottai - 627002

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Dr. A. MIchael J Leo, Assistant Professor, St.
Xavier's College of Education
(Autonomouis), Palayamkottai - 627002

16. Confirmatory Factor Analysis (CFA)

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Dr. A. MIchael J Leo, Assistant Professor, St.
Xavier's College of Education
(Autonomouis), Palayamkottai - 627002

17. EFA Vs. CFA
•EFA is a method for finding latent variables
in data, usually data sets with a lot of
variables.


•CFA is a method of confirming that certain
structures in the data are correct; often,
there is an hypothesized model due to
theory and to confirm it.
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Dr. A. MIchael J Leo, Assistant Professor, St.
Xavier's College of Education
(Autonomouis), Palayamkottai - 627002

18. EFA CFA
Before Knowing the Factor
Structure
After Knowing the Factor
Structure
Measurement Error is not
Captured
Measurement Error is
Captured and Estimated
with Model
For instruments that have
never been tested before.
For instruments that have
been tested before
When we translate an
existing instrument or Use
which was used in different
Sector or sample.
When it is Available
instantly as tailor made .
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Dr. A. MIchael J Leo, Assistant Professor, St.
Xavier's College of Education
(Autonomouis), Palayamkottai - 627002

19. Construct Validity
•Convergent Validity (CV)


•Discriminant Validity (DV)
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Dr. A. MIchael J Leo, Assistant Professor, St.
Xavier's College of Education
(Autonomouis), Palayamkottai - 627002

20. Convergent Validity (CV)
•Estimated by the Value of Average Variance
Extracted (AVE).

•AVE >=0.5 (Hair, Black, Babin & Anderson, 2009)

•Formula
Sum(Standardized Loading
2
)
AVE = ________________________
Number of Indicators

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Dr. A. MIchael J Leo, Assistant Professor, St.
Xavier's College of Education
(Autonomouis), Palayamkottai - 627002

21. Discriminant Validity (DV)


•DV =Square Root of (AVE)

•The DV should be Greater than the
correlation values among the corresponding
latent variables.

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Dr. A. MIchael J Leo, Assistant Professor, St.
Xavier's College of Education
(Autonomouis), Palayamkottai - 627002

22. Confirmatory Factor Analysis (CFA)
•Chi-Square Ratio=Chi-square value/ Degrees
of Freedom)
•The Ratio should be less than 2 or 3
•Then this is a better Model Fit with the Data
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Dr. A. MIchael J Leo, Assistant Professor, St.
Xavier's College of Education
(Autonomouis), Palayamkottai - 627002

23. Goodness of Fit Indices
•Comparative Fit Index (CFI)
•Norm Fit Index (NFI)
•Goodness-of-Fit Index (GFI)
•Adjusted Goodness-of-Fit Index (AGFI)
•These values is expected to be >=0.9
•Estimates: Standard Regression Weights(Beta)
Beta > 0.8 --- significant influence
0.8 > Beta > 0.5 --- moderate influence
Beta < 0.2 --- small influence

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Dr. A. MIchael J Leo, Assistant Professor, St.
Xavier's College of Education
(Autonomouis), Palayamkottai - 627002

24. Badness of Fit Indices
Root Mean Square Error
of Approximation
(RMSEA)


<0.8
Standardized Root Mean
Square Residual (SRMR)

Lesser the Value, the
Model is better fit
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Dr. A. MIchael J Leo, Assistant Professor, St.
Xavier's College of Education
(Autonomouis), Palayamkottai - 627002

25. Conclusion
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Dr. A. MIchael J Leo, Assistant Professor, St.
Xavier's College of Education
(Autonomouis), Palayamkottai - 627002

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Dr. A. MIchael J Leo, Assistant Professor, St.
Xavier's College of Education
(Autonomouis), Palayamkottai - 627002