1. SOC 681 – Causal Models with Directly Observed Variables
James G. Anderson, Ph.D.
Purdue University

2. Types of SEMs
Regression Models
Path Models
Recursive
Nonrecursive

3. Class Exercise: Example 7 SEMs with Directly Observed Variables
Felson and Bohrnstedt’s study of 209 girls from 6th through 8th grade
Variables
Academic: Perceived academic ability
Attract: Perceived attractiveness
GPA: Grade point average
Height: Deviation of height from the mean height
Weight: Weight adjusted for height
Rating: Rating of physical attractiveness

6. Assumptions
Relations among variables in the model are linear, additive and causal.
Curvilinear, multiplicative and interaction relations are excluded.
Variables not included in the model but subsumed under the residuals are assumed to be not correlated with the model variables.

7. Assumptions Variables are measured on an interval scale.
Variables are measured without error.

8. Objectives
Estimate the effect parameters (i.e., path coefficients). These parameters indicate the direct effects of a variable hypothesized as a cause of a variable taken as an effect.
Decompose the correlations between an exogenous and endogenous or two endogenous variables into direct and indirect effects.
Determine the goodness of fit of the model to the data (i.e., how well the model reproduces the observed covariances/correlations among the observed variable).

9. AMOS Input ASCII SPSS Microsoft Excel Microsoft Access
Microsoft FoxPro
dBase
Lotus

10. AMOS Output Path diagram
Structural equations effect coefficients, standard errors, t-scores, R2 values
Goodness of fit statistics
Direct and Indirect Effects
Modification Indices.

11. Model One

12. Decomposing the Effects of Variables on Achievement
Direct
Indirect
Total
Sex
-.03 -
FatherEd
.17
Ethnic
IndTrng
.23*
AStress
-.17*
ActMast
.02
SelfCon
.42*

13. Model Two

14. Goodness of Fit: Model 2 Chi-Square = 29.07 df = 15 p < 0.06
RMSEA = 0.086
GFI = 0.94
AGFI = 0.85
AIC = 67.82

15. Chi Square: 2 Best for models with N=75 to N=100
For N>100, chi square is almost always significant since the magnitude is affected by the sample size
Chi square is also affected by the size of correlations in the model: the larger the correlations, the poorer the fit

16. Chi Square to df Ratio: 2/df
There are no consistent standards for what is considered an acceptable model
Some authors suggest a ratio of 2 to 1
In general, a lower chi square to df ratio indicates a better fitting model

17. Root Mean Square Error of Approximation (RMSEA)
Value: [ (2/df-1)/(N-1) ]
If 2 < df for the model, RMSEA is set to 0
Good models have values of < .05; values of > .10 indicate a poor fit.

18. GFI and AGFI (LISREL measures)
Values close to .90 reflect a good fit.
These indices are affected by sample size and can be large for poorly specified models.
These are usually not the best measures to use.

19. Akaike Information Criterion (AIC)
Value: 2 + k(k-1) - 2(df)
where k= number of variables in the model
A better fit is indicated when AIC is smaller
Not standardized and not interpreted for a given model.
For two models estimated from the same data, the model with the smaller AIC is preferred.

20. Model Building Standardized Residuals ACH – Ethnic = 3.93
Modification Index ACH – Ethnic = 10.05

21. Model Three

22. Goodness of Fit: Model 3 Chi-Square = 16.51 df = 14 p < 0.32
RMSEA = 0.037
GFI = 0.96
AGFI = 0.90
AIC = 59.87

23. Comparing Models
Chi-Square Difference = df Difference = p < .0005
AIC Difference = 7.95

24. Difference in Chi Square
Value: X2diff = X2 model 1 -X2 model 2
DFdiff = DF model 1 –DFmodel 2

25. Decomposing the Effects of Variables on Achievement
Direct
Indirect
Total
Sex -
.09
FatherEd .-
.06
Ethnic
.29
.05
.34
IndTrng
.25
.04
AStress
-.14
-.03
-.17
ActMast
.13
SelfCon
.44

26. Class Exercise: Example 7 SEMs with Directly Observed Variables
Attach the data for female subjects from the Felson and Bohrnstedt study (SPSS file Fels_fem.sav)
Fit the non-recursive model
Delete the non-significant path between Attract and Academic and refit the model
Compare the chi square values and the AIC values for the two models

27. Class Exercise: Example 7 SEMs with Directly Observed Variables
Felson and Bohrnstedt’s study of 209 girls from 6th through 8th grade
Variables
Academic: Perceived academic ability
Attract: Perceived attractiveness
GPA: Grade point average
Height: Deviation of height from the mean height
Weight: Weight adjusted for height
Rating: Rating of physical attractiveness