1. G89.2247 Lecture 3 Review of mediation Moderation SEM Model notation
Estimating SEM models
G Lect 3

2. Review of Mediation We wish to “explain” modeled path c: with:
Total mediation models identify instruments for sophisticated structural equation models. X c e Y Y ey X M c' b eM a
G Lect 3

3. Nonrecursive models
All the models we have considered are recursive. The causal effects move from one side of the diagram to the other.
A Nonrecursive model has loops or feedback. X1 Y1 X2 Y2 e1 e2
G Lect 3

4. Example
Let X1 be college aspirations of parents of adolescent 1 and X2 be aspiration of parents of adolescent 2.
Youth 1 aspirations (Y1) are affected by their parents and their best friend (Y2), and Youth 2 has the reciprocal pattern.
This model is identified because it assumes that the effect of X1 on Y2 is completely mediated by Y1.
Special estimation methods are needed. OLS no longer works.
G Lect 3

5. Moderation
Baron and Kenny (1986) make it clear that mediation is not the only way to think of causal stages
A treatment Z may enable an effect of X on Y
For Z=1 X has effect on Y
For Z=0 X has no effect on Y
When effect of X varies with level of Z we say the effect is Moderated
SEM methods do not naturally incorporate moderation models
G Lect 3

6. Moderation, continued
In multiple regression we add nonlinear (e.g. multiplicative) terms to linear model
Covariance matrix is expanded
Distribution of sample covariance matrix is more complex
SEM ability to represent latent variables in interactions is limited
Easiest case is when moderator is discrete
G Lect 3

7. Path Diagrams of Moderation
Suppose that X is perceived efficacy of a participant and Y is a measure of influence at a later time. Suppose S is a measure of perceived status. Perceived status might moderate the effect of efficacy on influence.
Two ways to show this:
Equation: Y=b0+b1X+b2S+b3(X*S)+e
For S low:
For S high: X Y e
(+)
+ + X S Y e
G Lect 3

8. SEM and OLS Regression
SEM models and multiple regression often lead to the same results
When variables are all manifest
When models are recursive
The challenges of interpreting direct and indirect paths are the same in SEM and OLS multiple regression
SEM estimates parameters by fitting the covariance matrix of both IVs and DVs
G Lect 3

9. SEM Notation for LISREL (Joreskog)
Lisrel's notation is used by authors such as Bollen z1 b1 Y1 z2 g1 Y2 g2 X1
G Lect 3

10. SEM Notation for EQS (Bentler)
EQS does not name coefficients. It also does not distinguish between exogenous and endogenous variables. E2 V2 E3 V3 V1
G Lect 3

11. SEM Notation for AMOS (Arbuckle)
AMOS does not use syntax, and it has no formal equations. It is graphically based, with user-designed variables. E1
Fiz E2 Fa
Foo
G Lect 3

12. Matrix Notation for SEM
Consider LISREL notation for this model:
g11 z1 X1 Y1 b2 b1
g22 z2 X2 Y2
G Lect 3

13. More Matrix Notation
The matrix formulation also requires that the variance/covariance of X be specified
Sometimes F is used, sometimes SXX.
The variance/covariance of z is also specified
Conventionally this is called Y.
When designing structural models, the elements of F and Y can either be estimated or fixed to some (assumed) constants.
G Lect 3

14. Basic estimation strategy
Compute sample variance covariance matrix of all variables in model
Call this S
Determine which elements of model are fixed and which are to be estimated. Arrange the parameters to be estimated in vector q.
Depending on which values of q are assumed, the fitted covariance matrix S(q) has different values
Choose values of q that make the S and S(q) as close as possible according using a fitting rule
G Lect 3

15. Estimates Require Identified Model
An underidentified model is one that has more parameters than pieces of relevant information in S.
The model should always have
where t is the number of parameters, p is the number of Y variables and q is the number of X variables
Necessary but not sufficient condition
G Lect 3

16. Other identification rules
Recursive models will be identified
Bollen and others describe formal identification rules for nonrecursive models
Rules involve expressing parameters as a function of elements of S.
Informal evidence can be obtained from checking if estimation routine converges
However, a model may not converge because of empirical problems, or poor start values
G Lect 3

17. Review of Expectations
The multivariate expectations
Var(X + k*1) = S
Var(k* X) = k2S
Let CT be a matrix of constants. CT X=W are linear combinations of the X's.
Var(W) = CT Var(X) C = CT S C
This is a matrix
G Lect 3

18. Multivariate Expectations
In the multivariate case Var(X) is a matrix
V(X)=E[(X-m) (X-m)T]
G Lect 3

19. Expressing S(q) If Then We get S(q) by specifying the model details.
We also consider Cov(XY)
G Lect 3

20. Estimation Fitting Functions
ML minimizes
ULS minimizes
GLS minimizes
G Lect 3

21. Excel Example
G Lect 3

22. Choosing between methods
ML and GLS are scale free
Results in inches can be transformed to results in feet
ULS is not scale free
All methods are consistent
ML technically assumes multivariate normality, but it actually is related to GLS, which does not
Parameter estimates are less problematic than standard error estimates
G Lect 3