Bayesian Elastic-Net and Fused Lasso for Semiparametric Structural Equation Models: An Application in Understanding the Relationship Between Alcohol Morbidity and Other Substance Abuse Factors Among American Youth

In contemporary times, high-dimensional datasets have become increasingly prevalent, owing to the expansion and complexity of data collection facilitated by advancements in computer science, biology, and related fields. Analyzing such high-dimensional data poses distinct challenges compared to traditional data analysis, particularly in the realm of variable selection. Structural Equation Modeling (SEM) serves as a pivotal tool for scrutinizing the relationships between observable (manifest) variables and underlying (latent) variables. Traditionally, SEM primarily focuses on elucidating these relationships among latent variables. This paper proposes an extension of semiparametric structural equation modeling, which employs natural cubic splines to approximate nonlinear functional relationships. Moreover, we introduce priors based on Fused Lasso and Elastic Net to address correlations within both covariates and spline expansions. Through comprehensive simulation studies and real-world data analyses, we validate the efficacy of our approach. Our semiparametric structural equation models, enhanced with Bayesian fused Lasso and Bayesian elastic-net priors, consistently outperform conventional Bayesian Lasso models in both simulated and real-world datasets.