Multidimensional constructs and moderated linear and nonlinear factor analysis

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, I introduce a multidimensional MNLFA model that permits the moderation of item intercepts, loadings, residual variances, factor means, variances, and correlations across three or more latent factors. I describe efforts to implement the model using Bayesian methods through Stan and penalized maximum likelihood approaches to stabilize estimation and detect partial measurement non-invariance while preserving model interpretability. Closed-form analytic gradients of the likelihood, eliminating the need for costly numerical or MCMC-based approximations. 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.