Estimating Context Effects in Small Samples while Controlling for Covariates: An Optimally Regularized Bayesian Estimator for Multilevel Latent Variable Models

In this article, we extend the regularized Bayesian estimator of multilevel latent variable models (Dashuk et al., 2024) to improve the estimation of the between-group parameter βb in two-level latent variable models with covariates. Specifically, we allow for the inclusion of covariates and optimize the estimator’s accuracy in terms of the Mean Squared Error (MSE). Simulation results indicate that the regularized Bayesian estimator consistently outperforms both standard and transformed maximum likelihood (ML) estimators, especially in scenarios with small sample sizes and low intraclass correlations. While the estimator achieves lower Root Mean Square Error (RMSE) values and relative bias, it exhibited poorer coverage rates and less reliable standard errors compared to ML estimators. To address these limitations, we propose some strategies. Furthermore, a transformed ML estimator is utilized to enhance the accuracy of the estimation of the covariate parameter γ. Finally, we provide a step-by-step tutorial demonstrating how to apply the extended regularized Bayesian estimator to a real dataset using the MLOB package in R.