Structural equation
modeling:
Fit Analysis
Outline of the Presentation
•
•
•
•
Definitions and basics
Theory of implied covariance matrix
Implication of theory in our student evaluation model
R code (Lavaan)
• Degrees of freedom
• Interpretation of results
• Final discussion
Definition
•
Structural equation modeling is a multivariate
regression model which estimates causal relations
between variables.
Courtesy to Kiran Pedada and Jiyoon An
Fit Analysis
• What is fit Analysis?
• Fit refers to the ability of a model to reproduce the data (e.g.
variance-covariance matrix).
• A good-fitting model is one that is reasonably consistent with
the data
Courtesy to David Kenny
URL: http://davidakenny.net/cm/fit.htm
http://courses.ttu.edu/isqs5347-westfall/isqs5347.htm
Fit Analysis
• When data are produced by the specified SEM, the observed and
fitted covariance matrix will differ by chance alone.
• Simulation resolves the mystery!
http://courses.ttu.edu/isqs6348-westfall/images/6348/sim_cfa_r.txt
Courtesy to Dr. Westfall
Fit Analysis
• When data are produced by a model that is close to the specified
SEM(e.g, there are truly additional paths but their coefficients are
small), the observed and fitted covariance matrix may differ more than
can be explained by chance, but should still be close.
Courtesy to Dr. Westfall
General mathematical form
of latent variable model
•
Assumptions of the latent variable
model
•
The measurement model
•
Sample covariance
matrix
•
Definition of implied
covariance matrix
•
How to identify the
implied covariance matrix
•
Implied covariance matrix
•
Implied covariance matrix
in our case
•
Implied covariance matrix
•
Fit Analysis
•
• In this statistical hypothesis testing(Fit analysis) we would like to
see that our null hypothesis supported
• The likelihood ratio test is a test of full model versus restricted
model(null model).
• Full model is MVN with unrestricted covariance matrix
• Restricted model is MVN with restricted covariance matrix
• Restricted models always have lower likelihood.
• Chi square statistic = 2 x (difference between log likelihoods).
Dr. Westfall, ISQS5347 notes
Baseline model Chi square
• Full model is MVN with unrestricted covariance matrix, get a log
likelihood.
• Restricted model is MVN with restricted covariance matrix (this
time, with 0’s off the diagonal), get a log likelihood.
• As we all know, restricted models always have lower
likelihood.. The chi square statistics is 2 x(difference between log
likelihoods).
Dr. Westfall, ISQS5347 notes
• One of the key issues in calculating chi square is degree of freedom:
• In our case df= number of parameters in the unrestricted covariance
matrix (p(p+1)/2) minus number of free parameters to estimate in
the sem.
Dr. Westfall
• If the chi square test insignificant, then the difference between the
log likelihoods is explainable by chance alone. I.e., the difference is
within the realm of differences that are commonly seen when the
data are in fact simulated from a SEM.
Dr. Westfall
Other fit indices
•
The RMSEA Index
•
Discussion
• Why Chi square is so important?
• How degree of freedom affect the fitted model?