Regularized Joint Maximum Likelihood Estimation for Exploratory Multidimensional Item Response Theory Models

Exploratory multidimensional item response theory (MIRT), also known as exploratory item factor analysis, presents substantial computational and statistical challenges in large-scale applications, particularly when the latent space is high-dimensional (HiMIRT). Building on a previously proposed joint maximum likelihood estimation framework, this paper presents two advanced regularized MIRT methods: \textit{lasso HiMIRT} and \textit{card HiMIRT}. Both methods leverage the Minimization-Majorization (MM) algorithm to transform the likelihood problem into a least squares problem. \textit{lasso HiMIRT} incorporates an $\ell_1$ penalty within an alternating optimization–alternating direction method of multipliers (AO-ADMM) framework to achieve sparse loading matrix estimation. In contrast, \textit{card HiMIRT} imposes cardinality constraints to control the number of nonzero loadings in the model. Simulation studies show that both proposed methods accurately recover the true sparse structure of the loading matrix, effectively distinguishing between zero and nonzero elements. In addition to their precision, both approaches demonstrate strong computational efficiency. The proposed alternating optimization framework is also straightforward to implement and extend. The complementary strengths of the two approaches may benefit the practical applications of MIRT analysis. An empirical example based on data from the revised version of Eysenck’s Personality Questionnaire (EPQ-R) further illustrates their practical applicability.