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However, model equivalence is still very much an issue for two reasons. 4 An example of the generation of equivalent structural models applying the replacing rule to a symmetric focal block. V1 E1 Equivalent Model 5: V1 E1 Equivalent Model 4: 18  S. L. HERSHBERGER and G. A. 5 A proposed one common factor measurement model. general common factor. 5 that depicts a single common factor defined by five observed indicators. , higher than their correlation with the other three variables). In order to modify the model so as to accommodate the relatively high correlation between V1 and V2, any one of four mutually equivalent measurement models could be readily specified.

MARCOULIDES The Problem of Equivalent Structural Models  17 them may be replaced by a single directed path or a residual covariance. Importantly, the nonrecursive paths must be specified as equal due to the impossibility of uniquely identifying each path when both variables have the same predictors. The converse of this rule holds as well: Directed paths or residual covariances within a symmetric focal block may be replaced by equated nonrecursive paths (symbolized herein by ⇔). In Model 4, let V7 ⇔ V8 be a focal block where V7 → V8 = V7 ← V8; then V7 ⇔ V8 may be replaced by either V7 → V8, V7 ← V8, or E7 ↔ E8.

There can in fact be situations in which the degrees of freedom for a model are quite high and yet some of its parameters remain under-identified (Raykov & Marcoulides, 2006). Conversely, having negative degrees of freedom is a sufficient but not a necessary criterion for showing that a model is globally under-identified. Saturated models have as many parameters as there are nonredundant data elements and are commonly referred to as just-identified. Just-identified models are always identified in a trivial way.

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