Residuals and the Residual-Based Statistic for Testing Goodness of Fit of Structural Equation Models

The residuals obtained from fitting a structural equation model are crucial ingredients in obtaining chi-square goodness-of-fit statistics for the model. The authors present a didactic discussion of the residuals, obtaining a geometrical interpretation by recognizing the residuals as the result of oblique projections. This sheds light on the concept of degrees of freedom of the model. The authors use a simple example to illustrate the theory and also to provide simulations of residuals in three dimensions. They then explain the rationale behind the formula for the residual-based test statistic. The formula for the statistic is deduced using linear algebra and large-sample theory. Details are provided so that this material can be used in graduate instruction.