Single- and Multiple-Group Penalized Factor Analysis: A Trust-Region Algorithm Approach with Integrated Automatic Multiple Tuning Parameter Selection

Penalized factor analysis is an efficient technique that produces a factor loading matrix with many zero elements thanks to the introduction of sparsity-inducing penalties within the estimation process. However, sparse solutions and stable model selection procedures are only possible if the employed penalty is non-differentiable, which poses certain theoretical and computational challenges. This article proposes a general penalized likelihood-based estimation approach for single- and multiple-group factor analysis models. The framework builds upon differentiable approximations of non-differentiable penalties, a theoretically founded definition of degrees of freedom, and an algorithm with integrated automatic multiple tuning parameter selection that exploits second-order analytical derivative information. The proposed approach is evaluated in two simulation studies and illustrated using a real data set. All the necessary routines are integrated into the R package penfa. SUPPLEMENTARY INFORMATION: The online version contains supplementary material available at 10.1007/s11336-021-09751-8.