Multidimensional constructs and moderated linear and nonlinear factor analysis

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.