Can Ridge and Elastic Net Structural Equation Modeling be Used to Stabilize Parameter Estimates when Latent Factors are Correlated?

ABSTRACT

Multicollinearity between predictors is a common concern in SEM applications. As in linear regression models, high correlations between predictors can lead to unstable parameter estimates (i.e., large standard errors) and reduced statistical power. Regularized estimation methods, which have recently become available for SEMs, may provide more stable estimates in the presence of multicollinearity at the cost of a certain amount of bias in the estimated parameters. In a simulation study, we compared the performance of nonregularized SEM with Ridge and Elastic net regularized SEMs in the presence of strong multicollinearity. The results provide evidence that Ridge and Elastic net regularized SEMs provide more stable estimates and greater statistical power than nonregularized SEM. However, the biases from regularized estimation can result in increased Type I error rates. This phenomenon was more pronounced in Ridge than in Elastic net regularized SEMs. We discuss when the benefits can outweigh this cost.

KEYWORDS:

• Multicollinearity
• structural equation models
• regularization
• ridge
• penalized likelihood

Acknowledgments

We would like to thank Richard Rau for his valuable comments on an earlier version of this article.

Notes

¹ The shrinkage parameter is often denoted as $\lambda$. We deviate from this convention to avoid any confusion with the factor loading matrix $\mathrm{\Lambda }$ in the SEM context.

² Given the focus of the present article, we only considered regularization in the structural part of the model. Regularization in the measurement part (e.g., the factor loadings in $\mathrm{\Lambda }$) was addressed by Huang et al. (2017) and Scharf and Nestler (2019).

³ For the sake of transparency, we want to highlight the ways in which our simulation design deviated from Grewal et al. (2004): To reduce the complexity of the Results section, we left out a simulation condition with a different configuration of the structural coefficients (${\beta }_1={\beta }_2$). Further, we limited the number of different values for ${\phi }_{31}$ and ${\phi }_{32}$ and omitted an intermediate condition with a reliability of 0.8. We also only simulated an $R^2$ of 0.25 because we considered this condition to be the most realistic condition for typical psychological research questions.