Bayesian approach to structural equation models for ordered categorical and dichotomous data

Abstract

Structural equation modeling (SEM) is a statistical methodology that is commonly used to study the relationships between manifest variables and latent variables. In analysing ordered categorical and dichotomous data, the basic assumption in SEM that the variables come from a continuous normal distribution is clearly violated. A rigorous analysis that takes into account the discrete nature of the variables is therefore necessary. A better approach for assessing these kinds of discrete data is to treat them as observations that come from a hidden continuous normal distribution with a threshold specification. A censored normal distribution and truncated normal distribution, each includes interval, right and left where the later are with known parameters, are used to handle the problem of ordered categorical and dichotomous data in Bayesian non-linear SEMs. The truncated normal distribution is used to handle the problem of non-normal data (ordered categorical and dichotomous) in the covariates in the structural model. Two types of thresholds (having equal and unequal spaces) are used in this research. The Bayesian approach (Gibbs sampling method) is applied to estimate the parameters. SEM treats the latent variables as missing data, and imputes them as part of Markov chain Monte Carlo (MCMC) simulation results in the full posterior distribution using data augmentation. An example using simulation data, case study and bootstrapping method are presented to illustrate these methods. In addition to Bayesian estimation, this research provide the standard error estimates (SE), highest posterior density (HPD) intervals and a goodness-of-fit test using the Deviance Information Criterion (DIC) to compare with the proposed methods. Here, in terms of parameter estimation and goodness-of-fit statistics, it is found that the results with a censored normal distribution are better than the results with a truncated normal distribution, with equal and unequal spaces of thresholds. Furthermore, the results with unequal spaces of thresholds are less than the results of equal spaces of thresholds in the interval of the censored and truncated normal distributions, this is including the left censored and truncated normal distributions. The results of equal spaces of thresholds are less than the results of unequal spaces of thresholds in right censored and truncated normal distributions. In other cases, the results of bootstrapping method are better than the real data results in terms of SE and DIC. The results of convergence showed that dichotomous data needs more iterations to convergence than ordered categorical data.