Confirmatory Tetrad Analysis

A "tetrad" refers t o the difference i n the products of certain covariances (or correlations) among four random variables. A structural equation mode l often implies that some tetrads should be zero. These "vanishing tetrads" provide a means t o test structural equation models. In this paper we develop confirmatory tetrad analysis ( C T A ) . C T A applies a simultaneous test statistic for multiple vanishing tetrads developed b y Bollen (1990). The simultaneous test statistic is available in asymptotically distribution-free o r normal-distribution versions and applies to covariances o r t o correlations. We also offer new rules for determining the nonredundant vanishing tetrads implied by a model and develop a method to estimate the power of thestatistical test for vanishing tetrads. Testing vanishing tetrads provides a test for model fit that can lead t o results different f rom the usual likelihood-ratio ( L R ) test associated ~ ~ i t h the m a x i m u m likelihood methods that dominate the structural equation field. Also , the C T A technique applies t o some underidentified models. Furthermore, some models that are not nested according t o the traditional L R test are nested in terms of vanishing tetrads. Finally, C T A does not require numerical minimization and thus avoids the associated convergence problems that are present with other estimation approaches.