1. Structural Equation Modeling
Natasha Hudek and Stephanie Rattelade

2. Other Names for SEM Causal modeling Causal analysis
Simultaneous equation modeling
Analysis of covariance structures
Path analysis
Confirmatory factor analysis

3. Structural Equation Modeling
Confirmatory Approach
Combines Factor Analysis and multiple regression
Data = Model + Residual
Path Diagram – a visual representation of the variables and their relationships

4. Key Terms
Latent Variable – cannot be observed and is not measured directly
Observed Variable - measurable variables
Exogenous – causes changes in other variables
Endogenous – dependent and mediating variables
Parameters – the relationships between each variable to be estimated

5. Relationships between Variables
Association – a relationship between two variables that is nondirectional
Direct Effect – a directional relationship between two variables
Indirect Effect – effect of an exogenous variable on a endogenous variable through a mediating variable
Total effect – the sum of direct and indirect effects on a variable

6. Path Diagram 1 E1 E2 1 1
SAT
School Success
Income
Education 1 E3

7. Path Diagram Legend
Circles/Elipses: Latent variables/factors (not directly measured) and error terms
Rectangles/Squares: Observed variables (measured)
Lines indicate relationships between variables

8. Path Diagram Legend
Single arrow: directional relationship between an IV and DV
Double arrow: unanalysed relationship/covariance between 2 variables
Also represents variable’s variance
Stars indicate free parameters, numbers represet fixed parameters

9. Model Components
Measurement model – shows the relationships between observed and latent variables
Structural model – the relationships between latent variables
Recursive model – unidirectional links between variables
Nonrecursive model – reciprocal links between variables

10. Models
First-Order Factor Model
Second-Order Factor Model

11. When to Use SEM 1 + IVs and 1 + DVs
IVs and DVs can be continuous or discrete, latent or observed
Combines factor analysis and multiple regression
Must have a hypothesis, SEM is confirmatory, not exploratory

12. When to Use SEM Testing a model: How well does the model fit the data?
Testing a theory: Which model/theory fits the data better?
Variances and reliabilities:
Amount of variance in the DVs accounted for by the IVs
Reliability of observed/measured variables
Parameter estimates: What is the coefficient predicting one DV from one IV

13. When to Use SEM Mediating/moderating relationships
Group differences in model fit: Does the model fit 2 or more groups?
Differences across time: Longitudinal differences within and between people
Latent Growth Curve Modeling
Nested models: IVs predict DVs at various levels of multilevel models

14. Assumptions Large sample size No missing data Multivariate normality
No outliers
Linearity (among observed variables)
No multicollinearity and singularity
Residuals should be small and their covariances should be symmetrical

15. Note on Covariance Algebra
Covariance algebra can be used to solve SEM models
Each DV has it’s own equation
Several steps are required in the calculation of each covariance
For complex models, these calculations become time consuming

16. Step 1: Specification 1 E2 E1 1 1 SAT (V2) School Success (F1)
Income (V1)
Education (V3) 1 E3

17. Step 1: Specification
Relationships defined in the model are converted into equations/matrices to be estimated
Bentler-Weeks method uses every variable in the model as an IV or DV, and estimates:
Regression coefficients, and
Variances and covariances of IVs

18. Step 1: Specification Regression coefficients: h = bh + gx
Where, for example, h = a 3x1 vector, b = a 3x3 matrix, g = a 3x4 matrix, and x = a 4x1 vector, so
h = b h g x =
Ex: v2 = 0v2 + 0v3 + *f1 + 0v1 + 0e1 + 1e2 + 0e3, so
v2 =*f1 + e2

19. Step 1: Specification Variances and covariances of the IVs:
f = a 4x4 matrix
f =

20. Step 2: Estimation
Population parameters are estimated to produce a covariance matrix for the model
Start values are used to make initial guesses at the coefficients and variances to be estimated
For example:
= , = , =

21. Step 2: Estimation
Observed/measured variables are extracted from the full parameter matrices using a selection matrix (G):
For observed DVs:
Y = Gy*h =
For observed IVs:
X = Gx*x = v1
Rewriting the regression equation to express the DVs as a linear combination of IVs, we get:
h = (I – b)-1gx

22. Step 2: Estimation Using the formula
We get the covariance matrix between the DVs
For example: =

23. Step 2: Estimation Using the formula
We get the covariance matrix between the IVs and DVs
For example: =

24. Step 2: Estimation Using the formula
We get the covariance matrix between the IVs and
For example: =

25. Step 2: Estimation
We then combine , , and to get an initial covariance matrix =
This is what we get after one iteration
Iterations continue until the function converges, and a solution is reached
Our example took 12 iterations

26. Step 2: Estimation The final estimated parameters for the example are:
= , = , =
And the final estimated population covariance matrix is: =
The final residual matrix, where S is the sample covariance matrix, is:
S =

27. Step 2: Estimation 1 391.84 E1 E2 1 1 SAT (V2) School Success (F1)
20.49
2.56
Income (V1)
.35
Education (V3)
.14 1 E3

28. Step 3: Evaluation Model “fit” is evaluated using c2
In the example, convergence occurred at c2 = 0.47
Can also compute c2 from, c2 = i(n-1), where I is the minimum value obtained by convergence, and n is sample size
Sample size affects c2 value

29. Step 3: Evaluation To get degrees of freedom, So, df = 6 – 5 = 1
df = # data points - # parameters to be estimated
# data points = v(v+1)/2 = 3(3+1)/2 = 6, where v is # of measured variables
So, df = 6 – 5 = 1
c2 (1) = .47, p = .49

30. Step 3: Evaluation
Since our model fist the data well, we can look at individual relationships
Converting our 3 estimated regression parameters to z-scores:
Income  School Success = .347/.150 = 2.31, p = .02
School Success  SAT = /5.093 = 4.11, p = .000
School Success  Education = .143/.023 = 6.22, p = .000

31. Step 4: Modification
If you are happy with the evaluation results, you can leave your model as is.
If your c2 test proves significant, you may want to make revisions to your model and compare the old and new models
This is also done when testing a new theory versus a previous theory

32. SEM Programs
EQS
SAS CALIS
LISREL

33. AMOS

34. Creating a Path Diagram

35. Adding Parameters

36. Running the Output

37. Testing Models Default Model – model to be tested
Independence model – goodness-of-fit tested at 0
Saturated model – goodness-of-fit tested at 1

38. Unstandardized Outputs

39. Standardized Outputs

40. Outputs

41. Outputs

42. Variance-Covariance Matrix of Sample

43. Model Fit Summary

44. Interpretation
We can conclude that our hypothesized model fits the data
Next Steps:
Identify areas of misfit
Modify model relationships and retest