1. G89.2247 Lect 31 G89.2247 Lecture 3 SEM Model notation Review of mediation Estimating SEM models Moderation

2. G89.2247 Lect 32 SEM Notation for LISREL (Joreskog) Lisrel's notation is used by authors such as Bollen  Y2Y2 Y1Y1 X1X1    

3. G89.2247 Lect 33 SEM Notation for EQS (Bentler) EQS does not name coefficients. It also does not distinguish between exogenous and endogenous variables. EE V3V3 V2V2 V1V1 EE

4. G89.2247 Lect 34 SEM Notation for AMOS (Arbuckle) AMOS does not use syntax, and it has no formal equations. It is graphically based, with user-designed variables. EE Fa Fiz Foo EE

5. G89.2247 Lect 35 Revisiting Mediation We wish to “explain” modeled path c: with: Total mediation models identify instruments for sophisticated structural equation models. Y eyey X M c' b eMeM a X c e Y

6. G89.2247 Lect 36 Mediation: A theory approach Sometimes it is possible to argue on theoretical grounds that  Z is prior to X and Y  X is prior to Y  The effect of Z on Y is completely accounted for by the indirect path through X. This is an example of total mediation If   is fixed to zero, then Model 3 is no longer saturated.  Question of fit becomes informative  Total mediation requires strong theory

7. G89.2247 Lect 37 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. X1X1 Y1Y1 X2X2 Y2Y2 e1e1 e2e2

8. G89.2247 Lect 38 Equations Implied Assume all variables centered Predicting Y 1 : Y 1 = b 1 Y 2 + g 1 X 1 + e 1 Predicting Y 2 : Y 2 = b 2 Y 1 + g 2 X 2 + e 2 Note that the error terms cannot be assumed to be independent. There are no direct paths between X 1 and Y 2, or between X 2 and Y 1.

9. G89.2247 Lect 39 Example Let X 1 be college aspirations of parents of adolescent 1 and X 2 be aspiration of parents of adolescent 2. Youth 1 aspirations (Y 1 ) are affected by her parents and her best friend (Y 2 ), and Youth 2 has the reciprocal pattern. This model is identified because it assumes that the effect of X 1 on Y 2 is completely mediated by Y 1. Special estimation methods are needed.  OLS gives biased results.

10. G89.2247 Lect 310 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

11. G89.2247 Lect 311 Matrix Notation for SEM Consider LISREL notation for this model:  Y2Y2 Y1Y1 X2X2       X1X1 

12. G89.2247 Lect 312 More Matrix Notation The matrix formulation also requires that the variance/covariance of X be specified  Sometimes  is used, sometimes  XX. The variance/covariance of  is also specified  Conventionally this is called . When designing structural models, the elements of  and  can either be estimated or fixed to some (assumed) constants.

13. G89.2247 Lect 313 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 .  Depending on which values of  are assumed, the fitted covariance matrix  (  ) has different values Choose values of  that make the S and  as close as possible according using a fitting rule

14. G89.2247 Lect 314 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

15. G89.2247 Lect 315 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

16. G89.2247 Lect 316 Review of Expectations The multivariate expectations  Var( X + k* 1 ) =   Var(k* X ) = k 2  Let C T be a matrix of constants. C T X=W are linear combinations of the X's.  Var(W) = C T Var(X) C = C T  C This is a matrix

17. G89.2247 Lect 317 Multivariate Expectations In the multivariate case Var(X) is a matrix  V(X)=E[(X-  ) (X-  ) T ]

18. G89.2247 Lect 318 Expressing  If Then We get  by specifying the model details.  We also consider Cov(XY)

19. G89.2247 Lect 319 Estimation Fitting Functions ML minimizes ULS minimizes GLS minimizes

20. G89.2247 Lect 320 Excel Example

21. G89.2247 Lect 321 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

22. G89.2247 Lect 322 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

23. G89.2247 Lect 323 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

24. G89.2247 Lect 324 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=b 0 +b 1 X+b 2 S+b 3 (X*S)+e Path Diagrams of Moderation X S Y e For S low: For S high: XY e XY e (+) +