🤖 AI Summary
Existing multidimensional factor models struggle to accommodate the typical 3–5 dimensional latent constructs in psychometrics and lack a unified framework for modeling multiparameter moderation effects. This paper proposes a scalable penalized maximum likelihood estimation method applicable to arbitrarily many factors, enabling— for the first time—the joint estimation of linear and nonlinear moderation effects within high-dimensional models. By incorporating ridge, lasso, and alignment penalties, the approach simultaneously stabilizes parameter estimation, detects partial measurement noninvariance, and enhances interpretability. Leveraging closed-form analytical gradients, the method avoids computationally intensive numerical integration and MCMC sampling, substantially improving computational efficiency. Simulation and empirical studies demonstrate accurate recovery of complex moderation patterns. The proposed method provides a scalable, efficient, and robust new tool for measurement invariance research involving multidimensional constructs.
📝 Abstract
Multidimensional factor models with moderations on all model parameters have so far been limited to single-factor and two-factor models. This does not align well with existing psychological measures, which are commonly intended to assess 3-5 dimensions of a latent construct. In this paper, we introduce a penalized maximum likelihood approach for multidimensional MNLFA that permits moderation of item intercepts, loadings, residual variances, factor means, variances, and correlations across three or more latent factors. Our approach incorporates ridge, lasso, and alignment penalties to stabilize estimation and detect partial measurement non-invariance while preserving model interpretability. We derive closed-form analytic gradients of the likelihood, eliminating the need for costly numerical or MCMC-based approximations, and demonstrate how this dramatically improves computational efficiency. Through simulation and application, we illustrate that penalized MNLFA recovers complex moderation patterns in multidimensional constructs and provides a scalable alternative to Bayesian implementations. We conclude by discussing the theoretical implications of penalization for measurement invariance, computational considerations, and future directions for extending the framework to categorical indicators, longitudinal data, and applied research contexts.