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 models M 1 and M 2 when fitting the more restricted model M 0 . 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 of M 1 and M 2 (which characterizes model M 12 ), is a necessary and sufficient condition for the local equivalence of M 1 and M 2 . This condition is satisfied, in practice, when the analysis dealing with the fitting of M 0 , predicts that the decreases in the chi-square goodness-of-fit statistic for the fitting of M 1 or M 2 , or M 12 are all equal for any set of sample data, except on differences due to rounding errors.