Slide 10.1 Structural Equation Models MathematicalMarketing Chapter 10 Structural Equation Models In This Chapter We Will Cover The theme of this chapter is causation. We have already looked at one class of causal models, confirmatory factor analysis. We will now look at a more general formulation including  Path analysis  All-y models  Nonrecursion  Path models with latent variables  Second order factor models  Models that include means

Slide 10.2 Structural Equation Models MathematicalMarketing Key Terminology  Endogenous  Exogenous  Path analysis  Causal model  Structural equation model  Covariance structure model

Slide 10.3 Structural Equation Models MathematicalMarketing The Basic SEM Model vector of p endogeneous variables regression coefficients for endogeneous variables on other endogeneous varibles regression coefficients for endogeneous variables on exogeneous variables specification error exogeneous variables

Slide 10.4 Structural Equation Models MathematicalMarketing Some Basic Assumptions E(y) = 0 E(x) = 0 Cov(x,  ) = 0 V(x) = E(xx) =  V(  ) = E(  ) = . Some Definitions

Slide 10.5 Structural Equation Models MathematicalMarketing Reduced Form Coefficients vs. Structural Coefficients

Slide 10.6 Structural Equation Models MathematicalMarketing What Are the Model's Implications for V(y)? E(yy) = GE(xx)G + E(ee)

Slide 10.7 Structural Equation Models MathematicalMarketing Proceeding With the V(y) E(yy) = GE(xx)G + E(ee) G = (I – B) -1  E(xx) = V(x) =  e = (I – B) -1  V(  ) = 

Slide 10.8 Structural Equation Models MathematicalMarketing Another Important Piece: Cov(x, y) e = (I – B) -1 

Slide 10.9 Structural Equation Models MathematicalMarketing The Structure of the Covariance Matrix for All the Variables

Slide Structural Equation Models MathematicalMarketing A Simple Causal Model Intention to Purchase y1y1 Purchasing Behavior y2y2 Perceived Costx2x2 Attractiveness of Product x1x1 Description Variable

Slide Structural Equation Models MathematicalMarketing Graphical Representation of Path Models x2x2 y1y1 y2y2 x1x1  21  11  12  Boxes are manifest variables  Circles are latent variables  Unlabeled arrows are error  Labeled single headed arrows are causal paths  Double headed arrows are covariances  21

Slide Structural Equation Models MathematicalMarketing The Equations for the Sample Model

Slide Structural Equation Models MathematicalMarketing We Do Not Really Need x and y Variables Rewriting the model slightly y = By +  x +  x = 0y + Ix + 0 Now define so the model is

Slide Structural Equation Models MathematicalMarketing We Can Also Consolidate the Covariance Matrices Now for our model We have the following covariance matrices for the exogenous factors

Slide Structural Equation Models MathematicalMarketing So We Don't Need So Many Matrices  Instead of having x and y we can get by with one set of variables: z  Instead of having B and  we can get by with just G  Instead of having  and  we can get by with just A

Slide Structural Equation Models MathematicalMarketing In What Sense Are These Causal Models? y3y3 y2y2 y1y1 y2y2 y1y1 y3y3 The "same" arrow is missing in both.

Slide Structural Equation Models MathematicalMarketing Both Models Have 1 DF and That DF Implies the Same Restriction Both causal diagrams require only that the partial covariance  23·1 = 0 where  23·1 is the Cov(e 2, e 3 ) and e 2 and e 3 are defined as the errors in y 2 = y 1 + e 2 and y 3 = y 1 + e 3 Assuming that this constraint is met, that just means we have failed to reject H o. Of course that doesn't mean we have proven it.

Slide Structural Equation Models MathematicalMarketing Regression Is a Special Case of a Causal Model x2x2 y1y1 x1x1  21  11  12 x3x3  13  32  31 What are the degrees of freedom for this model? Can we reject the causal hypothesis?

Slide Structural Equation Models MathematicalMarketing Multivariate Regression Is Also a Special Case x2x2 y1y1 x1x1  21  11  12 x3x3  23  32  31 y2y2  22  21  13  12 The same questions apply here

Slide Structural Equation Models MathematicalMarketing Recursive Systems A recursive system is characterized by V(  ) =  diagonal, and by the fact that it is possible to arrange the y variables so that B is lower (or upper) triangular x2x2 y1y1 x1x1 y2y2 x2x2 y1y1 x1x1 y2y2 Two Nonrecursive Examples

Slide Structural Equation Models MathematicalMarketing Structural Equation Models with Latent Variables y =  y  +  x =  x  +   = B  +  +  Measurement models Structural equation models

Slide Structural Equation Models MathematicalMarketing Some of the Assumptions of the Model Cov ( ,  ) = 0 Cov ( ,  ) = 0 Cov ( ,  ) = 0 Cov ( , ,  ) = 0 Diag (B) = 0 | I – B|  0

Slide Structural Equation Models MathematicalMarketing Naming Some More Matrices y =  y  +  x =  x  +   = B  +  +  V(  ) =   V(  ) =  V(  ) =  V(  ) =  

Slide Structural Equation Models MathematicalMarketing An Example Longitudinal Model with Latent Variables y1y 11  21 y2y2 y3y 22  32 y4y4 y5y 33  43 y6y6 y7y 44 y8y8

Slide Structural Equation Models MathematicalMarketing The Matrices for the Model

Slide Structural Equation Models MathematicalMarketing Variance Matrices for Exogenous Factors The Matrices for the Model

Slide Structural Equation Models MathematicalMarketing The Second Order Factor Analysis Model y =  y  +   =   +  V(  ) =   V(  ) =  V(  ) = 

Slide Structural Equation Models MathematicalMarketing Path Diagram of Second Order Factor Analysis y2y2 y1y1 y3y 11 22 y5y5 y4y4 y6y6 33 44 y7y7 y8y8 1  41  31  21

Slide Structural Equation Models MathematicalMarketing Models with Structured Means x = x +  x  +  y = y +  y  +   =  + B  +   +  E(x) = x +  x  E(y) = y +  y (I – B) -1 (  +  ) E(  ) = (I - B) -1 (  +  ) x measurement model. Includes x 0 y measurement model SEM  = E(  )

Slide Structural Equation Models MathematicalMarketing Details on the Model 1 = 1  0 + 0

Slide Structural Equation Models MathematicalMarketing A Sequence of Hypotheses for Multiple Group Analysis