Asymptomatic Independence and Separability in Convariance Structure Models: Implications for Specification Error, Power, and Model Modification

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Abstract
This article presents a didactic discussion on the role of asymptotically independent test statistics and separable hypotheses as they pertain to issues of specification error, power, and model modification in the covariance structure modeling framework. Specifically, it is shown that when restricting two parameter estimates on the basis of the multivariate Wald test, the condition of asymptotic independence is necessary but not sufficient for the univariate Wald test statistics to sum to the multivariate Wald test. Instead, what is required is mutual asymptotic independence (MAI) among the univariate tests. This result generalizes to sets of multivariate tests as well. When MA1 is lacking, hypotheses can exhibit transitive relationships. It is also shown that the pattern of zero and non-zero elements of the covariance matrix of the estimates are indicative of mutually asymptotically independent test statistics, separable and transitive hypotheses. The concepts of MAI, separability, and transitivity serve as an explanatory framework for how specification errors are propagated through systems of equations and how power analyses are differentially affected by specification errors of the same magnitude. A small population study supports the major findings of this article. The question of univariate versus multivariate sequential model modification is also addressed. We argue that multivariate sequential model modification strategies do not take into account the typical lack of MA1 thus inadvertently misleading substantive investigators. Instead, a prudent approach favors univariate sequential model modification.