Confirmatory Tetrad Analysis

A refers to the difference in the products of certain covariances (or correlations) among four random variables. A structural equation model often implies that some should be zero. These tetrads provide a means to test structural equation models. In this paper we develop confirmatory tetrad analysis (CTA). CTA applies a simultaneous test statistic for multiple vanishing developed by Bollen (1990). The simultaneous test statistic is available in asymptotically distribution-free or normal-distribution versions and applies to covariances or to correlations. We also offer new rules for deternmining the nonredundant vanishing implied by a model and develop a method to estimate the power of the statistical test for vanishing tetrads. Testing vanishing provides a test for model fit that can lead to results different from the usual likelihood-ratio (LR) test associated with the maximum likelihood methods that dominate the structural equation field. Also, the CTA technique applies to some underidentified models. Furthzermore, some models that are not nested according to the traditional LR test are nested in terms of vanishing tetrads. Finally, CTA does not require numerical minimization and thus avoids the associated convergence problems that are present with other estimation approaches.