G89 2247 Lecture 3 Review of mediation Moderation SEM Model notation Estimating SEM models G89 2247 Lect 3 1 Review of Mediation We wish to explain modeled path c e X with c Y c X a M Y ey b eM Total mediation models identify instruments for sophisticated structural equation models G89 2247 Lect 3 2 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 e1 X2 Y2 e2 G89 2247 Lect 3 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 4 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 5 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 6 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 For S low Two ways to show this X Y X Y e For S high e X Y S e Equation Y b0 b1X b2S b3 X S e G89 2247 Lect 3 7 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 8 SEM Notation for LISREL Joreskog Lisrel s notation is used by authors such as Bollen X1 Y1 Y2 Y BY X 1 Y1 0 0 Y1 1 1 Y 0 Y X 1 2 1 2 2 2 G89 2247 Lect 3 9 SEM Notation for EQS Bentler EQS does not name coefficients It also does not distinguish between exogenous and endogenous variables E V2 V3 E V1 G89 2247 Lect 3 10 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 Fiz Fa E Foo G89 2247 Lect 3 11 Matrix Notation for SEM Consider LISREL notation for this model X1 X2 Y1 Y2 Y BY X Y1 0 Y 2 2 1 Y1 11 0 X 1 1 0 Y2 0 22 X 2 2 G89 2247 Lect 3 12 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 G89 2247 Lect 3 13 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 G89 2247 Lect 3 14 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 t p q 1 p q 2 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 15 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 16 Review of Expectations The multivariate expectations Var X k 1 Var k X k2 Let CT be a matrix of constants CT X W are linear combinations of the X s Var W CT Var X C CT C This is a matrix G89 2247 Lect 3 17 Multivariate Expectations In the multivariate case Var X is a matrix V X E X X T 12 13 14 2 21 2 23 24 2 31 32 3 34 2 41 42 43 4 2 1 G89 2247 Lect 3 18 Expressing If Y BY X Then Var Y I B 1 XX T I B 1 T We get by specifying the model details We also consider Cov XY G89 2247 Lect 3 19 Estimation Fitting Functions ML minimizes F ln ln S tr S 1 ULS minimizes F tr S S GLS minimizes F tr S S 1 S S 1 G89 2247 Lect 3 20 Excel Example G89 2247 Lect 3 21 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 22