Equivalent models in covariance structure analysis

Defining equivalent models as those that reproduce the same set of covariance matrices, necessary and sufficient conditions are stated for the local equivalence of two expanded identified modelsM₁ andM₂ when fitting the more restricted modelM₀. Assuming several regularity conditions, the rank deficiency of the Jacobian matrix, composed of derivatives of the covariance elements with respect to the union of the free parameters ofM₁ andM₂ (which characterizes modelM₁₂), is a necessary and sufficient condition for the local equivalence ofM₁ andM₂. This condition is satisfied, in practice, when the analysis dealing with the fitting ofM₀, predicts that the decreases in the chi-square goodness-of-fit statistic for the fitting ofM₁ orM₂, orM₁₂ are all equal for any set of sample data, except on differences due to rounding errors.This research was supported by the Foundation of Social-Cultural Sciences which is subsidized by the Dutch Scientific Organization (N.W.O.) under project number 500-278-003. The author wishes to thank Anne Boomsma, Ivo Molenaar, Albert Satorra, and Tom Snijders for their stimulating and crucial comments during the research, and the Editor, Paul Bekker, Henk Broer, and anonymous reviewers for their helpful suggestions.