Croon's Corrections for Estimating Single-Level Latent Interactions

• In Event: Composite/Factor Scores Analyses With Bias Correction: Good Alternatives to Full Structural Equation Modeling?

Abstract

Purpose: The primary purpose of this study was to extend and probe the performance of Croon’s method of estimation to latent interactions in (1) a moderated mediation model involving two two-way latent interactions and (2) in a model with a three-way interaction and two two-way latent interactions. A secondary purpose of this study was to compare currently available estimation approaches to the newly developed Croon method with complex models containing latent interactions and limited sample sizes. Criteria for evaluating estimator performance included convergence failure rate, bias, and efficiency.

Theoretical Framework: Interactions between a focal independent variable and another variable (i.e., moderator) have long been an important consideration for educational researchers with interactions often occurring between latent variables (e.g., Aguinis et al., 2005; Aiken & West, 1991). Unfortunately, the computational complexity of estimating latent interactions, has limited model complexity. Recent advancements have shown promise including Bayesian estimation (Asparouhov & Muthen, 2020) but performance is still problematic with limited sample sizes and it is unclear if currently recommended estimators work well in models with multiple latent and higher order interactions. We extend the Croon method to latent interactions and compare its performance with a variety of estimators to address this problem and gap.

Methods: The Croon method is a limited information sequential estimation approach that separates estimation of the measurement and structural models. Its distinguishing feature is a Croon’s method of moments correction to adjust estimates in the structural model to account for latent variable unreliability (Croon, 2002). After detailing the extension of Croon’s method to latent interactions, we probe its performance with other estimators in a simulation with data based on a moderated mediation model involving two two-way latent interactions and another model with a three-way and two two-way latent interactions. We tracked convergence rate and compared average absolute bias and the standard deviation of coefficient estimates from the newly developed Croon method, direct maximum likelihood estimation, an uncorrected factor score approach, and Bayesian estimation with various limited sample sizes in both models.

Results: The Croon method consistently converged when estimating both models and across sample sizes. It produced the least amount of average absolute bias while being nearly as efficient as the Bayesian approach. Conversely, maximum likelihood estimation was found to be impractical across models and sample sizes due to extreme computational times and the uncorrected factor score approach produced substantially more bias than either the Croon or Bayesian estimators.

Significance: This work established the Croon method for estimating latent interactions and probed its performance. Results suggest it performs well even compared to the highly recommended Bayesian estimator. Improved estimation of latent interactions expands the scope of models available for consideration and increases the feasibility of studies that consider latent interactions but have limited sample sizes. We have also increased understanding of absolute and relative estimator performance in complex models with latent interactions and limited sample sizes. This information can better guide researchers in the selection of an estimation approach for their specific analysis.

Author

• Kyle T. Cox, University of North Carolina - Charlotte