Can There Be Infinitely Many Models Equivalent to a Given Covariance Structure Model?

The problem of equivalent models has plagued researchers since the earliest developmental stages of structural equation modeling (SEM; Marcoulides & Schumacker, 1996). Equivalent models are those that provide the same sets of statistical fit indexes (e.g., chi-square and p values) as a hypothesized model but may imply very different substantive interpretations of the data (Stelzl, 1986). As a consequence, equivalent models represent a very serious threat to behavioral and social theory development and construct validation using SEM. In fact, unless the model equivalence problem is addressed, any originally considered model—regardless of how well it fits the data—remains only one possible means of its explanation (e.g., MacCallum, Wegener, Uchino, & Fabrigar, 1993). Although the problem of equivalent models has received considerable methodological attention for more than a decade (e.g., Bollen, 1989; Breckler, 1990; Hayduk, 1996; Hershberger, 1994; Joreskog & Sorbom, 1993; Lee & STRUCTURAL EQUATION MODELING, 8(1), 142–149 Copyright © 2001, Lawrence Erlbaum Associates, Inc.