Statistical Equivalence, Semantic Equivalence, Eliminative Induction, and the Raykov-Marcoulides Proof of Infinite Equivalence

Statistically equivalent models produce the same range of moment matrices over the domain of their parameter spaces. Raykov and Marcoulides (2001) proposed a proof that leads to the conclusion that all structural equation (SE) models with certain minimal components have infinitely many statistically equivalent models. A variation on their proof covers an even broader class of models. This conclusion has important implications for the application of at least one notion of eliminative induction to structural equation modeling (SEM). Normally, assertions of statistical equivalence imply that the models differ in meaning, giving statistical equivalence its interest. Consequently, a particular complex causal structure provides a counterexample to the proposed proof. This counterexample suggests that a successful proof may require more detailed attention to the concept of semantic equivalence as characterized by different substantive implications. A formal account of semantic equivalence rests on translation between SE models and a model-neutral descriptive language.