Updated 10-14-2002

Confirmatory Factor Analysis - CFA


Same form of the model as Exploratory Factor analysis


\begin{displaymath}\underline{X} = \underline{\mu} + \underline{\Lambda} \underline{f} + \underline{\epsilon} \end{displaymath}

but in CFA, the $Var(\underline{f})=\underline{\Phi}$ is not necessarily constrained to be the identity matrix, AND constraints can be placed on the elements of $\underline{\Lambda}$, AND $Var{\underline{\epsilon}} = \underline{\Psi}$ is not necessarily constrained to be diagonal (i.e. the errors may be correlated).

As with EFA, we are interested in fitting the model covariance matrix ${\underline{\Sigma}}(\underline{\theta})$ to the sample covariance matrix.


\begin{displaymath}{\underline{\Sigma}}(\underline{\theta}) = {\underline{\Lambda}} \Phi {\underline{\Lambda}} + {\underline{\Psi}} \end{displaymath}

where $\underline{\theta}$ represents all the free parameters in $\underline{\Lambda}$, $\underline{\Phi}$, and $\underline{\Psi}$.

Maximum likelihood assuming normality for the data will be used to estimate $\hat{\underline{\theta}}$.

The degrees of freedom for the confirmatory factor analysis model equals p*(p+1)/2 - number of free parameters in $\underline{\theta}$.

Comparison of EFA to CFA - EXAMPLE

Exploratory Factor analysis as a precursor to Confirmatory factor analysis, E-mail from Bengt Muthen on SEMNET here.
More on the subject can be found at HTTP://bama.ua.edu/cgi-bin/wa?A1=ind0010&L=semnet

Requirements for CFA Model Identification.
(pp. 188-192 of Maruyama talks about identification for SEM in general)

Necessary conditions:

If the CFA model has all measurement errors uncorrelated, (i.e. $\Psi$ is diagonal) and satisfies both of the necessary conditions above, then the model is identified if

If the CFA model has some measurement errors correlated, defining rules for identifiability is difficult. One common recommendation is to the let the software crank it out. AMOS can give a message that the model is not identified (based on empirical considerations). This use is not fool proof though, it is possible the software can miss. (see Bollen 1989 and Rigdon 1995 for more rules).

Model fitting (pp.196-201 in Maruyama)

Let ${\underline{\Sigma}}$ represent the true covariance structure of the data. We want to find a model ${\underline{\Sigma}}(\theta)$that describes ${\underline{\Sigma}}$.

AMOS LAB - using the visual perception/ verbal ability data fit a two factor confirmatory factor analysis model.

degrees of freedom should be 21 - 13 = 8. There are 13 parameters being estimated.....

\begin{displaymath}\Lambda = \left( \begin{array}{c} 1 \\ \lambda_{21} \\ \lambd... ...ay} \begin{array}{c} \phi_{12}\\ \phi_{22} \end{array} \right) \end{displaymath} \begin{displaymath}\Psi = \left( \begin{array}{c} \psi_1 \\ 0 \\ 0 \\ 0 \\ 0 \\ ... ...n{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ \psi_6 \end{array}\right) \end{displaymath}

NESTED Models (pp. 235-238 in Maruyama)

Modification Indices

Sample Size and Normality

Chi-squared test of model fit and standard errors for factor loadings are based on asymptotic theory

For another good reference for non-normality, check out http://www.utexas.edu/cc/fa qs/stat/general/gen33.html Handling non-normal data in SEM and http://www.utexas.edu/cc/faqs/ stat/amos/amos7.html Handling non-normal data using AMOS

Here is a .pdf file containing notes I made outlining an article by Hu, Bentler (1995) ``Evaluating Model Fit'' in SEM: Concepts, Issues, and Applications, edited by Hoyle, RH, Thousand Oaks, Calif., Sage Pub, p. 76-99.

A newer reference by Hu and Bentler is: Hu, L-T., & Bentler, P.M. (1999). "Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives", Structural Equation Modeling, 6, 1-55.

Also check out: West S., Finch, J., and Curran, P.J. "Structural Equation Models with nonnormal variables: Problems and remedies" in SEM: Concepts, Issues, and Applications. Edited by Hoyle RH. Thousand Oaks, CA, Sage Pub. pp 56-75. Note to MW: make notes from this article and put here

Other Fit Indices and Testing procedures (We will touch on these topics later)