ABSTRACT
For many years, structural equation models have been fitted to summary statistics, primarily covariances, but sometimes to the means as well. Beginning in the 1990s, programs such as Mx and Amos have shown the advantages of fitting models to raw data, and it is extensions of this method that form the main focus of this chapter. A frequently asked question is: “How does one fit structural equations models to the raw data by maximum likelihood?” In some ways, this is a strange question because it is a simpler question to answer than one that asks about the origin of the formula used for fitting models to covariance matrices. Therefore, this chapter begins with an elementary introduction to maximum likelihood (ML), including the concepts of individual fit and the multivariate normal distribution. The second section discusses various alternative measures of individual fit, suitable for use when some of the data are missing. The third section considers moderator variables as a potential source of non-normality. If these moderators have been measured, it is possible to explicitly model their effects. For the case of binary moderators, the model may be specified as a two-group structural equation model, but continuous moderators require an extension to ML analysis of raw data such that there is a different model for every subject in the sample. Finally, although individual-fit statistics can be useful for detecting outliers or mixture distributions and for judging the value of adding moderating variables, heterogeneity may not always be directly related to an observed moderator variable. Formal methods for detecting “latent” heterogeneity require the application of finite-mixture distributions, which are described in the fourth section.
