Structural Equation Modeling: Applications Using Mplus

Appendix 1.B Maximum Likelihood Function for SEM

In SEM model estimation, attention is directed to the sample distribution of the observed variance/covariance matrix S. If a random sample is selected from a multivariate normal population, the likelihood of finding a sample with variance/covariance matrix S is given by the Wishart distribution (Wishart, 1928):

(1.33)

where S is the sample variance/covariance matrix, is the population variance/covariance matrix, n = N − 1 (where N is sample size), K is the number of variables, and is the gamma function. Note that all the terms in Equation (1.33), except those involving , are constant. Since we are only interested in maximizing the function rather than calculating the precise value of the function, all the constant terms in Equation (1.33) can be combined into one constant term C, thus the equation can be simplified to:

(1.34)

For a model that fits data perfectly, . As such, the ratio of the Wishart function of the specified model to ...