The document outlines the differences between exploratory factor analysis (EFA) and confirmatory factor analysis (CFA), emphasizing CFA's role in theory testing and measurement validation. It details the assumptions, parameters, and identification criteria for CFA models, along with various model fit indices and the implications of sample size on chi-square measures. The document also discusses parameter restrictions, computation methods, and the significance of modification indices in improving model fit.

1. Confirmatory Factor
Analysis
Eduard Ponarin
Boris Sokolov
HSE, St. Petersburg
19.11.2013

2. EFA vs CFA
• Exploratory Factor Analysis:
preliminary exploration of data (data-
driven)
• Confirmatory Factor Analysis: test of
theory against data (theory-driven)

3. Why we need CFA?
• Empirical test for a theory
• Operationalization of a theory: do our
indicators actually measure our
constructs? (quality of our
questionnaire)
• Measurement part of structural model
(Instead of aggregated indices)

4. Theory Testing
• Specification of an initial model based
on theoretical considerations
• Estimation of the model: does it fit
data well?
• Modification of the model if needed.
Search for the best specification
• A necessary requirement: the
modified model should have not only
a good fit but also an appropriate
substantive interpretation

5. Assumptions of CFA
• Cov(A,e) = 0 (the latent variable is
not correlated with the measurement
error)
• E(e) = 0 (the expected values of
random measurement errors are
equal to zero)
• Multivariate normality
• Linearity

6. Parameters in CFA Model
Known Information (input matrix):
• Indicator variances and covariances
Unknown (freely estimated) parameters:
• Factor Loadings
• Factor Variances/Covariances
• Error Variances/Covariances
• Degrees of Freedoms = number of known
elements of the input matrix – number of
unknown parameters
• DF = (K*(K + 1))/2 – T where K is a number
of indicators and T is a number of free
parameters

7. Identification
• A model is identified when the sample
variance and covariance include enough
information (variances and covariances) to
estimate all free model parameters
• Under-Identified model: less
variances/covariances than unknown free
parameters (DF < 0)
• Just-identified model: (DF = 0)
• Over-Identified model: more
variances/covariances than unknown free
parameters (DF > 0)

8. Restrictions on Parameters
• (!) The more free parameters the less
probability that the model will be
identified
• Fixed parameters – parameters which
are fixed to a certain constant
• Constrained parameters – parameters
which are set to be equal

9. Nested models
• Congeneric
• Tau equivalent: equal factor loadings
• Parallel: equal factor loadings, equal
measurement errors
• Strictly parallel: equal factor loadings,
equal measurement errors, equal
intercepts

10. Some Typical Restrictions
• Set factor loading for the first
indicator to zero
• Set variance of the latent variable to
one

11. Computation of Model
Parameters
• Let assume factor model with three indicators X1, X2,
X3
• Factor loading for the first indicator is calculated as
follows:
• L1=Cov(X1,X2)*Cov(X1,X3)/Cov(X2,X3)
• Measurement error variance for the first indicator is
calculated as follows:
Var(e1) = Var(X1) – L1²
• Expected variance of the indicator is equal to:
VarExp(X1) = L1² + Var(e1)
• Expected Covariance between two indicators is equal
to product of their respective factor loadings
(unstandardized):
CovExp(X1,X2) = L1*L2

12. Model Fit
• Global Fit: differences between
expected and observed covariances
• (!) Fit Indices are informative only
when the model is over-identified
• A “good model fit” only indicates that
the model is plausible (neither that it
is ‘true’ nor that it explains a large
proportion of the covariance)

13. Goodness of Fit
• Chi-Square Difference Test
• Standardized Root Mean Square (SRMR): from 0 to
1. SRMR less than 0.08 indicates an acceptable fit.
• Root Mean Square Error of Approximation
(RMSEA): from 0 to infinity (but rarely exceeds 1).
RMSEA less than 0.05 indicates an acceptable fit
• Comparative Fit Index (CFI): from 0 to 1. CFI more
than 0.9 (0.95) indicates a good fit
• Tucker-Lewis Index (TFL). TFL close to1 (>0.9 or
0.95) indicates a well fitting model.
• Various Information Criteria (AIC, BIC, etc)

14. Chi-Square
• Difference between observed and
expected covariances
• Chi-Square goodness-of-fit measure is
sensitive to sample size
• When the sample size is small there is
a possibility of Type I error (that is,
rejecting the model that fits data well)
• When the sample size is large there is
a possibility of Type II error (that is,
accepting ‘false’ model)

15. Modification Indices
• MI is the amount of chi-square which
will drop if the parameter is estimated
as part of the model.
• MI more than 3.84 (5 or 10) are
reasonable.
• Error variances/covariances
• Factor variances/covariances
• Cross-loadings

16. Thank you for attention!