Structural equation modeling (SEM) is a statistical methodology commonly used in the social and behavioral sciences due to its ability to model complex systems of human behavior while allowing for the use of latent variables and variables measured with error. SEM differs markedly from other statistical methods due to its modeling of the covariance matrix of the observed variables as opposed to the individual observations themselves as done in many statistical methods. This difference is evident in how residual analysis is conducted. In many common statistical methods residual analysis consists of graphical displays of the residuals, residual-based model diagnostics, and residual-based hypothesis tests to assess model assumptions and detect potential outliers and influential observations. Though a number of diagnostics have been developed to assess the overall adequacy of a proposed SEM model and several simulation studies have assessed the effects of model misspecification, assumption violations, and outliers/influential observations, the use of residual analysis similar to that commonly employed in most statistical methods to assess the adequacy of a model has been largely neglected.

The goal of this dissertation is to further the use of residual analysis in SEM. First, the finite sample and asymptotic properties of a class of residual estimators that are weighted functions of the observed variables are derived. These properties are then assessed through the use of a simulation study. Second, the residuals constructed using the proposed class of residual estimators are examined for their ability to detect outliers and influential observations. These applications extend the use of residual plots and Cook's distance to the SEM framework. The utility of these proposed extensions are then evaluated through the use of two examples. Theoretical results indicate that the optimal estimator from the class of proposed estimators depends on the criterion used to evaluate the estimators. Empirical results from the two examples indicate the utility of the proposed residual plots and extension of Cook's distance to detect outliers and influential observations. Thus, this dissertation provides the basis for residual analysis in SEM.