An Empirical Bayesian Approach to Misspecified Covariance Structures

The analysis of covariance structures has been an important topic in psychometrics and latent variable modeling. A covariance structure is a model for the covariance matrix of observed manifest variables. It is derived from hypothesized linear relationships among the manifest variables and hypothesized unobserved latent variables. The traditional approach to covariance structures has been successful when the covariance structure is correctly specified, i.e., when the population covariance matrix satisfies the given covariance structure. However, in reality, covariance structures never hold exactly in the population as the hypotheses behind them are only approximations to the truth. Consequently it is necessary to model misspecification when covariance structures are analyzed. The traditional approach, nevertheless, only acknowledges and accounts for the effect of misspecification by post hoc modifications of the original approach to correctly specified covariance structures, and does not actively model the process that may have lead to misspecification.

In this dissertation, we present a new approach to misspecified covariance structures in which the systematic error, identified as the process behind misspecification, is explicitly modeled along with the sampling error as a stochastic quantity with a distribution, and the inverse sample size for this distribution, as an unknown parameter to be estimated, gives a measure of misspecification. Analytical properties of the maximum beta likelihood (MBL) procedure implied by this approach and its limit, the maximum inverted Wishart likelihood (MIWL) procedure, are investigated and several connections with the traditional approach are found. Computer programs that give numerical implementations of these procedures are provided. Asymptotic sampling distributions of estimators given by the above two procedures are derived under different replication frameworks with a much weaker assumption than the usually invoked Pitman drift assumption. Sampling experiments are conducted to validate the asymptotic sampling distributions and to demonstrate the importance to account for the variations in the parameter estimates due to systematic error.