1. Structural Equation Modeling

2. Structural Equation Modeling
Diagramming P. 374, Box 11.1—
Rules for Causal Diagrams
Exogenous vs. Endogenous Variables
Direct and Indirect Effects
Residual Variables
Observed Latent Residual, uncorrelated

3. Structural Equation Modeling
Measurement Models

4. Structural Equation Modeling
Structural Models

5. Structural Equation Modeling
Simultaneous Models

6. Structural Equation Modeling
Path Analysis
Path Analysis estimates effects of variables in a causal system.
It starts with structural Equation—a mathematical equation representing the structure of variables’ relationships to each other.
Education
PaEduc
Well-being
Income

7. Structural Equation Modeling
Path Analysis
Regress endogenous variables on their predictors
Variables are stated in terms of Z-scores
Standardized coefficients are the path coefficients
Path coefficient from the residual to a variable is √1 – R2 (the unexplained variation)
Squaring the path from a residual to a variable gives 1 – R2 % variance not explained by the model
Education
PaEduc
Well-being
Income

8. Structural Equation Modeling
Path Analysis
Path coefficients also account for your correlation matrix.
Decomposition gives you parts of the correlation between variables
Correlation between two variables is the additive value of the paths from one to the other. Along the way, paths are multiplied if a variable(s) intervenes along the way. Intuitively, this makes sense. If X Y Z, and the two coefficients are .5 each, the correlation of X and Z is This makes sense because if X goes up one standard deviation, Y goes up .5. Z will go up .5 for every one unit increase in Y, meaning that it will go up .25 for a .5 unit increase in Y, or a one unit increase in X. (Like gears turning in sequence).
Education
PaEduc
Well-being
Income

9. Structural Equation Modeling
Path Analysis
P. 384 for mathematical way to decompose
But really, all you need to do is trace paths. P. 386 for rules (let’s go through these)
If you specify a potential path as zero, you can test to see if the correlations between variables as hypothesized in that model match the real correlations
Education
PaEduc
Well-being
Income

10. Structural Equation Modeling
Confirmatory Factor Analysis
Same thing we proposed last week w/EFA and Cronbach’s Alpha
But now let’s assume the factor is standardized with a variance of 1, where an indicator’s loading equals 1—this is actually a way to create a scale for your factor.
Variance of X will equal correlation with the factor plus error.
The correlation of a pair of observed variables loading on a factor is the product of their standardized factor loadings.
Factor 1 X1 X3 X2 e1 e2 e3

11. Structural Equation Modeling
Confirmatory Factor Analysis
MLE replicates the covariance matrix by generating “parameters” (really statistics) that come as close as possible to the observed covariance matrix (see Vogt).
Weights are generated. They have unstandardized coefficient interpretations and do not represent the correlations with the factor. The biggest is not necessarily the best.
We’ll discuss
loadings later.
Factor 1 X1 X3 X2 e1 e2 e3

12. Structural Equation Modeling
Confirmatory Factor Analysis
Next, we should test the fit of the model.
This test compares the generated predicted covariance matrix with the real covariance matrix in the sample data.
The “fit function” multiplied by N-1 is distributed as a 2 with degrees of freedom = [k(k+1)/2] – t, where t is the number of independent parameters.
Some do not use 2 as a test statistic--think of 2 as equal to (S – E), where S is the sample covariance matrix and E is the expected from MLE.
Factor 1 X1 X3 X2 e1 e2 e3

13. Structural Equation Modeling
Confirmatory Factor Analysis
Some do not use 2 as a test statistic--think of 2 as equal to (S – E), where S is the sample covariance matrix and E is the expected from MLE.
Rejecting 2 has a “badness of fit” interpretation: Null: S = E
Alternative: S  E
p > .05 means fail to reject the null, the model has a good fit.
Factor 1 X1 X3 X2 e1 e2 e3

14. Structural Equation Modeling
Confirmatory Factor Analysis
Larger samples produce greater chance of rejecting the null, saying the fit is bad.
Therefore, some use the ratio of 2 /df and try to get this ratio under 2.
2 = 0 would imply perfect fit.
Factor 1 X1 X3 X2 e1 e2 e3

15. Structural Equation Modeling
Confirmatory Factor Analysis
Other measures of fit that don’t depend on sample size:
GFI: Analogous to R2, Fit of this model versus fit of no model (where all parameters equal zero). You try for .95 or higher.
GFI = [1 – (unexplained variability/total variability)]
AGFI: GFI adjusted for model complexity. More complex models reduce GFI more.
Factor 1 X1 X3 X2 e1 e2 e3

16. Structural Equation Modeling
Confirmatory Factor Analysis
Parameters
When model fit is good, you can begin focusing on “parameters.” Each has a standard error associated with it.
The AMOS output refers to critical ratios. A critical ratio is the parameter divided by its standard error. (a lot like a z test) If your critical ratio is 1.96 or larger, your parameter is significant.
Factor 1 X1 X3 X2 e1 e2 e3

17. Structural Equation Modeling
Confirmatory Factor Analysis
Loadings--Check to see if your loadings are equal.
Completely standardizing the model gives correlations and standardized coefficients, allowing you to compare them with each other on strength of association with the factor.
The square of the loading plus the square of the effect of the error term equals 1. All variation is coming from two sources.
Factor 1 X1 X3 X2 e1 e2 e3

18. Structural Equation Modeling
Confirmatory Factor Analysis
Remember the concept of scaling with variations in item weights? SEM allows for “automatic” weighting.
Factor 1 X1 X3 X2 e1 e2 e3

19. Structural Equation Modeling
In SEM, interpret path coefficients the way you would with OLS regression (or path analysis), which would give the same results.
Testing whether one model is better than another is simply a 2 test with degrees of freedom of the difference in df between the two models.
Null: Models are the same
Alternative: Models are different
Factor 1 X1 X3 X2 e1 e2 e3

20. Structural Equation Modeling
Testing whether one model is better than another is simply a 2 test with degrees of freedom of the difference in df between the two models.
Null: Models are the same
Alternative: Models are different
If 2 is not significant, the models are equivalent. Therefore, use the more restricted model with higher degrees of freedom.
If 2 is significant, the parameters cannot be the same between models. No constraint should be added.
Factor 1 X1 X3 X2 e1 e2 e3

21. Structural Equation Modeling
If 2 is not significant, the models are equivalent. Therefore, use the more restricted model with higher degrees of freedom.
If 2 is significant, the parameters cannot be the same between models. No constraint should be added.
This indicates a way to test whether two parameters are equal—just constrain them to be equal and compare the fit.
Ways to test whether parameters are zero—just constrain them to be zero and compare the fit.
Factor 1 X1 X3 X2 e1 e2 e3