An Adaptive Bayesian Lasso Approach with Spike-and-Slab Priors to Identify Multiple Linear and Nonlinear Effects in Structural Equation Models

Abstract

In applied research, such as with motivation theories, typically many variables are theoretically implied predictors of an outcome and several interactions are assumed (e.g., Watt, 2004). However, estimation problems that might arise when several interaction and/or quadratic effects are analyzed simultaneously have not been investigated because simulation studies on interaction effects in the structural equation modeling framework have mainly focused on small models that contain single interaction effects. In this article, we show that traditional approaches can provide estimates with low accuracy when complex models are estimated. We introduce an adaptive Bayesian lasso approach with spike-and-slab priors that overcomes this problem. Using a complex model in a simulation study, we show that the parameter estimates of the proposed approach are more accurate in situations with high multicollinearity or low reliability compared with a standard Bayesian lasso approach and typical frequentist approaches (i.e., unconstrained product indicator approach and latent moderated structures approach).

Keywords:

• Adaptive lasso
• continuous mixture
• interaction effect
• multicollinearity
• quadratic effect
• regularization
• reliability
• spike-and-slab priors

Notes

¹ The supplement can be downloaded from the first author’s personal website www.holger-brandts-methods.com.

² False constraints in the structural model (i.e., omitting relevant nonlinear effects) can result in spurious estimates for the remaining parameters and may not show up in model fit (Gerhard, Büchner, Klein, & Schermelleh-Engel, 2014; Gerhard et al., 2014; Klein et al., 2009). From this perspective, it is beneficial to include these nonlinear terms in this model.

³ Simulation results not reported here indicated that the performance of the aBSS-lasso did not depend on the prior specification with Half Cauchy or alternatively with Inverse Gamma distributions.

⁴ Additional simulation results not reported here indicated that the horseshoe prior has very similar characteristics to the aBSS-lasso with regard to bias and accuracy; however, convergence rates were lower with an average of 74%. They improved with sample size from 69% for to 80% for .

Additional information

Funding

This work was supported by the Deutsche Forschungsgemeinschaft [KE 1664/1-2].