Alleviating Estimation Problems in Small Sample Structural Equation Modeling-A Comparison of Constrained Maximum Likelihood, Bayesian Estimation, and Fixed Reliability Approaches

Small sample structural equation modeling (SEM) may exhibit serious estimation problems, such as failure to converge, inadmissible solutions, and unstable parameter estimates. A vast literature has compared the performance of different solutions for small sample SEM in contrast to unconstrained maximum likelihood (ML) estimation. Less is known, however, on the gains and pitfalls of different solutions in contrast to each other. Focusing on three current solutions-constrained ML, Bayesian methods using Markov chain Monte Carlo techniques, and fixed reliability single indicator (SI) approaches-we bridge this gap. When doing so, we evaluate the potential and boundaries of different parameterizations, constraints, and weakly informative prior distributions for improving the quality of the estimation procedure and stabilizing parameter estimates. The performance of all approaches is compared in a simulation study. Under conditions with low reliabilities, Bayesian methods without additional prior information by far outperform constrained ML in terms of accuracy of parameter estimates as well as the worst-performing fixed reliability SI approach and do not perform worse than the best-performing fixed reliability SI approach. Under conditions with high reliabilities, constrained ML shows good performance. Both constrained ML and Bayesian methods exhibit conservative to acceptable Type I error rates. Fixed reliability SI approaches are prone to undercoverage and severe inflation of Type I error rates. Stabilizing effects on Bayesian parameter estimates can be achieved even with mildly incorrect prior information. In an empirical example, we illustrate the practical importance of carefully choosing the method of analysis for small sample SEM. Translational Abstract Structural equation modeling (SEM) is one of the most popular analysis tools in the social and behavioral sciences. Under small samples, however, SEM may exhibit serious estimation problems, such as failure to converge, inadmissible solutions such as negative variance estimates, and unstable parameter estimates. A vast literature has compared the performance of different solutions for small sample SEM in contrast to customary maximum likelihood (ML) estimation. Less is known, however, on the gains and pitfalls of different solutions in contrast to each other. Focusing on three current solutions-constrained ML restricting parameters to admissible values, Bayesian estimation techniques, and model simplification using fixed reliability single indicator (SI) approaches-we bridge this gap. When doing so, we evaluate the potential and boundaries of different parameterizations, constraints, and weakly informative prior distributions for improving the quality of parameter estimates and inference. The performance of all approaches is compared in a simulation study. Under conditions with low reliabilities, in terms of accuracy of parameter estimates, Bayesian methods without additional prior information by far outperform constrained ML as well as the worst-performing fixed reliability SI approach and do not perform worse than the best-performing fixed reliability SI approach. Under conditions with high reliabilities, constrained ML shows good performance. Both constrained ML and Bayesian methods exhibit conservative to acceptable Type I error rates. Fixed reliability SI approaches are prone to severe inflation of Type I error rates. Stabilizing effects on Bayesian parameter estimates can be achieved even for mildly incorrect prior information. In an empirical example, we illustrate the practical importance of carefully choosing themethod of analysis for small sample SEM.