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

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 G89.2247 Lect 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 G89.2247 Lect 3

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. G89.2247 Lect 3

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 G89.2247 Lect 3

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 G89.2247 Lect 3

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 G89.2247 Lect 3

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 G89.2247 Lect 3

SEM Notation for LISREL (Joreskog) Lisrel's notation is used by authors such as Bollen z1 b1 Y1 z2 g1 Y2 g2 X1 G89.2247 Lect 3

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 G89.2247 Lect 3

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 G89.2247 Lect 3

Matrix Notation for SEM Consider LISREL notation for this model: g11 z1 X1 Y1 b2 b1 g22 z2 X2 Y2 G89.2247 Lect 3

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. G89.2247 Lect 3

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 G89.2247 Lect 3

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 G89.2247 Lect 3

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 G89.2247 Lect 3

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 G89.2247 Lect 3

Multivariate Expectations In the multivariate case Var(X) is a matrix V(X)=E[(X-m) (X-m)T] G89.2247 Lect 3

Expressing S(q) If Then We get S(q) by specifying the model details. We also consider Cov(XY) G89.2247 Lect 3

Estimation Fitting Functions ML minimizes ULS minimizes GLS minimizes G89.2247 Lect 3

Excel Example G89.2247 Lect 3

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 G89.2247 Lect 3