A tutorial on Bayesian structural equation modelling: Principles and applications

This paper explores the utilisation of Bayesian structural equation modelling (BSEM) in psychology, highlighting its advantages over frequentist methods for handling complex models and small sample sizes. Basic concepts and fundamental issues relevant to BSEM are introduced, such as prior setting, model convergence, and model fit evaluation and so on. The paper also provides illustrative examples of commonly employed BSEMs, including confirmatory factor analysis (CFA) models, mediation models and multigroup CFA models, accompanied by empirical data and computer codes to facilitate implementation. Our goal is to provide researchers with novel ideas for empirical research and equip them to overcome challenges inherent to traditional methods. As BSEM continues to gain traction in various fields, we anticipate its development will feature improved methods, techniques and reporting standards.

What Is the Maximum Likelihood Estimate When the Initial Solution to the Optimization Problem Is Inadmissible? The Case of Negatively Estimated Variances

The default procedures of the software programs Mplus and lavaan tend to yield an inadmissible solution (also called a Heywood case) when the sample is small or the parameter is close to the boundary of the parameter space. In factor models, a negatively estimated variance does often occur. One strategy to deal with this is fixing the variance to zero and then estimating the model again in order to obtain the estimates of the remaining model parameters. In the present article, we present one possible approach for justifying this strategy. Specifically, using a simple one-factor model as an example, we show that the maximum likelihood (ML) estimate of the variance of the latent factor is zero when the initial solution to the optimization problem (i.e., the solution provided by the default procedure) is a negative value. The basis of our argument is the very definition of ML estimation, which requires that the log-likelihood be maximized over the parameter space. We present the results of a small simulation study, which was conducted to evaluate the proposed ML procedure and compare it with Mplus’ default procedure. We found that the proposed ML procedure increased estimation accuracy compared to Mplus’ procedure, rendering the ML procedure an attractive option to deal with inadmissible solutions.

Editorial of the Psych Special Issue “Computational Aspects, Statistical Algorithms and Software in Psychometrics”

Statistical software in psychometrics has made tremendous progress in providing open source solutions (e [...]

Robust Chi-Square in Extreme and Boundary Conditions: Comments on Jak et al. (2021)

In this article we describe a modification of the robust chi-square test of fit that yields more accurate type I error rates when the estimated model is at the boundary of the admissible space.

Using the Effective Sample Size as the Stopping Criterion in Markov Chain Monte Carlo with the Bayes Module in Mplus

Bayesian modeling using Markov chain Monte Carlo (MCMC) estimation requires researchers to decide not only whether estimation has converged but also whether the Bayesian estimates are well-approximated by summary statistics from the chain. On the contrary, software such as the Bayes module in Mplus, which helps researchers check whether convergence has been achieved by comparing the potential scale reduction (PSR) with a prespecified maximum PSR, the size of the MCMC error or, equivalently, the effective sample size (ESS), is not monitored. Zitzmann and Hecht (2019) proposed a method that can be used to check whether a minimum ESS has been reached in Mplus. In this article, we evaluated this method with a computer simulation. Specifically, we fit a multilevel structural equation model to a large number of simulated data sets and compared different prespecified minimum ESS values with the actual (empirical) ESS values. The empirical values were approximately equal to or larger than the prespecified minimum ones, thus indicating the validity of the method.