1. Structural Equation Modeling
Karl L. Wuensch
Dept of Psychology
East Carolina University

2. Nomenclature AKA Two sets of variables “causal modeling.”
“analysis of covariance structure.”
Two sets of variables
Indicators – measured (observed, manifest) variables – diagramed within rectangles
Latent variables – factors – diagramed within ellipses

3. Causal Language
Indicators and latent variables may be classified as “independent” or “dependent”
Even if no variables are manipulated
Based on the causal model being tested.
In the diagram, indicators and latent variables may be connected by arrows
One-headed = unidirectional causal flow
Two-headed = direction not specified

4. Paths “Dependent” variables have one-headed arrows pointing to them.
“Independent” variables do not.
“Dependent” variables also have residuals (are not perfectly predicted by the independents)
Called errors (e) for observed variables
Disturbances (d) for latent variables

5. Two Models
Measurement Model – how the measured variables are related to the latent variables.
Structural Model – how the latent variables are related to each other.

6. Two Variance Covariance Matrices
The sample matrix – computed from the sample data.
The estimated population matrix – estimated from the model.
2 test of null that the model fits the data well.
More useful are goodness of fit estimators

7. Sample Size Need at least 200 cases even for a simple model.
Rule of thumb: at least 10 cases per estimated parameter.

8. Assumptions & Problems
Multivariate normality.
Linear relationships
But can include polynomial components
A singular matrix may crash the program
Multicollinearity can be a problem

9. Simple Example from T&F
The “Independent” Variables are shaded.

10. Regression Parameters
Regression coefficients for the paths
May be “fixed” to value 0 (no path) or 1
Or estimated from the data ().

11. Variance/Covariance Parameters
Variances/Covariances of the “independent” variables
May be estimated
or fixed, to 1 or
to the variance of a “marker” measured variable (set to 1 the path to the marker).
Variances for “dependent” latent variables are usually fixed to the variance of one of the measured variables (set to 1 the path to that measured variable).

12. Model Identification
A model is “identified” if there is a unique solution for each of the estimated parameters.
Determine the number of input data points (values in the sample variance/covariance matrix).
This is
Where m = number of measured variables.

13. Model Identification
For T&F’s simple model, 5(6)/2 = 15 data points.

14. Model Identification
If the number of data points = the number of parameters to be estimated, the model is “just identified,” or “saturated,” and the fit will be perfect.
If there are fewer data points than parameters to be estimated, the model is “under identified” and the analysis is kaput.

15. The “Over Identified” Model
The number of input data points exceeds the number of parameters to be estimated.
This is the desired situation.
For T&F’s simple model, count the number of asterisks in the diagram. I count 11.
15 input data points, 11 parameters to estimate  we have an over identified model.

16. Eleven Parameters (*)
The “Independent” Variables are shaded.

17. Identification of the Measurement Model Should Be OK if
Only one latent variable, at least three indicators, errors not correlated.
Two or more latent variables, each has at least three indicators, errors not correlated, each indicator loads on only one latent variable, latent variables are allowed to covary.

18. Identification of the Measurement Model Should Be OK if
Two or more latent variables, one has only two indicators, errors are not correlated, each indicator loads on only one latent variable, none of the latent variables has a variance or covariance of zero.

19. Identification of the Structural Model May Be OK if
None of the latent DVs predicts another latent DV,
or if one does, it is recursive (unidirectional) and the disturbances are not correlated
If there are nonrecursive relationships (an arrow from A to B and from B to A), hire an expert in SEM.

20. Error in Identification
If there is a problem, the software will throw an error.
The software may suggest a way to reach identification.
You must tinker with the model to make it identified.

21. Estimation Maximum Likelihood most common
An iterative procedure used to maximize fit between the sample var/cov matrix and the estimated population var/cov matrix.
Generalized Least Squares estimation has also fared well in Monte Carlo comparisons of techniques.

22. Modifying and Comparing Models
May simplify a model by deleting one or more parameters in it.
The simplified model is nested with the previous model.
Calculate difference 2 = difference between the two model’s Chi-Squares.
df = number of parameters deleted.

23. LM and Wald Tests Lagrange Multiplier Test Wald Test
Would fit be improved by estimating a parameter that is currently fixed?
Wald Test
Would fixing this parameter significantly reduce the fit?
Available in SAS Calis, not in Amos

24. Reliability of Measured Variables
Assume that the variance in the measured variable is due to variance in the latent variable (the “true scores”) + random error.
Reliability = true variance divided by (true and error variance).
Thus, estimated reliability = the r2 between measured variable and latent variable.