Structural Equation Modeling: Applications Using Mplus

Appendix 1.A Expressing Variances and Covariances Among Observed Variables as Functions of Model Parameters

Let us denote the population variance/covariance matrix of variables y and x, then

(1.22)

where the diagonal elements are variances of the variables y and x, respectively; and the off-diagonal elements are covariances among y and x. In SEM it is hypothesized that the population variance/covariance matrix of y and x can be expressed as a function of the model parameters , that is:

(1.23)

where is called the model implied variance/covariance matrix.

Based on the three basic SEM equations [Equation (1.1)], we can derive that can be expressed as functions of the parameters in the eight fundamental SEM matrices. Let us start with the variance/covariance matrix of y, then the variance/covariance matrix of x and the variance/covariance matrix of y and x, and then finally assemble them ...