Changing a Causal Hypothesis without Changing the Fit: some Rules for Generating Equivalent Path Models

Since computer programs have been available for estimating and testing linear causal models (e.g. LISREL by Jöreskog & Sörbom) these models have been used increasingly in the behavioral sciences. This paper discusses the problem that very different causal structures may fit the same set of data equally well. Equivalence of models is defined as undecidability in principle and four rules are presented showing how equivalent models may be generated. Rules I and II refer to inversions in the causal order of variables and Rules III and IV to replacements of paths by the assumption of correlated residuals. Three examples are given to illustrate how the rules are applied.