Posterior predictive checks of tetrad subsets for covariance structures of measurement models

Confirmatory tetrad analysis is based on tetrads that are assumed to be 0 or to "vanish" under an assumed model to test the goodness of fit of a model or the relative fit of nested models. A strength of a tetrad-based approach to model fit assessment over standard methods is that researchers can evaluate the plausibility of a theoretical model structure when estimation or identification is problematic. Previous research has developed confirmatory tetrad analysis for covariance structures using asymptotic and bootstrap sampling distributions for omnibus test statistics for all vanishing tetrads. We extend confirmatory tetrad analysis in 2 ways. First, we develop an approach to confirmatory tetrad analysis based on posterior predictive checks for Bayesian probability models using tetrads to construct discrepancy measures. Second, we show how the discrepancy measure for all vanishing tetrads can be additively decomposed into measures for subsets of tetrads using a typology of tetrads proposed by Kenny (1979). We show that posterior predictive checks using a hierarchy of subsets of tetrads from this typology can be effectively used to identify potential and specific sources for lack of fit. We demonstrate the utility of this approach with classic data sets from the literature and with simulated data.