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. We 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. We use a simple example to illustrate the theory and also to provide simulations of residuals in three dimensions. We 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.