Bayesian diagnostics of transformation structural equation models

In the behavioral, social, psychological, and the medical sciences, the most widely used models in assessing latent variables are the structural equation models (SEMs). However, most of the existing statistical methods for analyzing SEMs have been developed for normally distributed data. Transformation SEMs are useful tools for tackling the non-normality of multidimensional data and simultaneously revealing the interrelationships among latent variables. The main objective of this paper is to develop a Bayesian diagnostic procedure for transformation SEMs. The first- and second-order local inference measures are established with the objective functions that are defined based on the logarithm of Bayes Factor. Markov chain Monte Carlo (MCMC) methods with the Bayesian P-splines approach are developed to compute the local influence measures and to estimate nonparametric transformations, latent variables, and unknown parameters. Compared with conventional maximum likelihood-based diagnostic procedures, the Bayesian diagnostic approach could detect outliers and/or influential points in the observed data, as well as conduct model comparison and sensitivity analysis through various perturbations of the data, sampling distributions, or prior distributions of the model parameters. Simulation studies reveal the empirical performance of the proposed Bayesian diagnostic procedure. An actual data set that is extracted from a study based on the Hong Kong Diabetes Registry is used to illustrate the application of our methodology.