Structural Models and the Art of Approximation

Structural equation models have provided a seemingly rigorous method for investigating causal relations in nonexperimental data in the presence of measurement error or multiple measures of putative causes or effects. Methods have been developed for fitting these very complex models globally and obtaining global fit statistics or global measures of their approximation to sample data. Structural equation models are idealizations that can serve only as approximations to real multivariate data. Further, these models are multidimensional, and the approximation is itself multidimensional. Tests of “significance” and global indices of approximation do not provide an adequate basis for judging the acceptability of the approximation. Standard applications of structural models use a composite of two models—a measurement (path) model and a path (causal) model. Separate analyses of the measurement model and the path model provide an informed judgment, whereas the composite global analysis can easily yield unreasonable conclusions. Separating the component models enables a careful assessment of the actual constraints implied by the path model, using recently developed methods. An empirical example shows how the conventional global treatment yields unacceptable conclusions.