A semiparametric Bayesian approach for structural equation models

Abstract

In the development of structural equation models (SEMs), observed variables are usually assumed to be normally distributed. However, this assumption is likely to be violated in many practical researches. As the non-normality of observed variables in an SEM can be obtained from either non-normal latent variables or non-normal residuals or both, semiparametric modeling with unknown distribution of latent variables or unknown distribution of residuals is needed. In this article, we find that an SEM becomes nonidentifiable when both the latent variable distribution and the residual distribution are unknown. Hence, it is impossible to estimate reliably both the latent variable distribution and the residual distribution without parametric assumptions on one or the other. We also find that the residuals in the measurement equation are more sensitive to the normality assumption than the latent variables, and the negative impact on the estimation of parameters and distributions due to the non-normality of residuals is more serious. Therefore, when there is no prior knowledge about parametric distributions for either the latent variables or the residuals, we recommend making parametric assumption on latent variables, and modeling residuals nonparametrically. We propose a semiparametric Bayesian approach using the truncated Dirichlet process with a stick breaking prior to tackle the non-normality of residuals in the measurement equation. Simulation studies and a real data analysis demonstrate our findings, and reveal the empirical performance of the proposed methodology. A free WinBUGS code to perform the analysis is available in Supporting Information.

A Bayesian Modeling Approach for Generalized Semiparametric Structural Equation Models

“…As pointed out by Song, Pan, Kwok, Vandenput, Ohlsson, and Leung (2010), for SEMs with continuous latent variables, the Bayesian estimation is robust to misspecification of the distribution of explanatory latent variables but sensitive to that of random errors (or response variables). Semiparametric modeling approaches such as Dirichlet process with the stick-breaking prior (Yang & Dunson, 2010; Yang, Dunson, & Baird, 2010; Song et al, 2010) and non-parametric transformation (Song & Lu, 2012) can be used to handle non-normal random errors or response variables. For SEMs with discrete latent variables, latent class modeling technique (Nylund & Muthén, 2007; Pan, Song, & Ip, 2013) can be employed to handle categorical latent variables.…”

Bayesian Gaussian Copula Factor Models for Mixed Data

“…A number of authors have proposed mixtures of factor models (Chen et al, 2010;Ghahramani and Beal, 2000). Song et al (2010) instead allow flexible error distributions in Eq. (1.1).…”

Why Ordinal Variables Can (Almost) Always Be Treated as Continuous Variables: Clarifying Assumptions of Robust Continuous and Ordinal Factor Analysis Estimation Methods

“…It should be noted that there is active research in factor analysis for continuous variables with non-normally distributed factors or residuals that can be skewed, bounded, or are mixtures of distributions (Song et al, 2010;Kelava and Brandt, 2014;Zhang et al, 2014;Asparouhov and Muthén, 2016;Lin et al, 2016;Revuelta et al, 2020). Using these more complex distributions would reduce the degree of distributional misspecification in the factor model.…”