A penalized likelihood-based framework for single and multiple-group factor analysis models

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. Penalized models are generally less prone to instability in the estimation process and are easier to interpret and generalize than their unpenalized counterparts. However, sparse solutions and stable model selection procedures are only possible if the employed penalty is singular (non-differentiable) at the origin, which poses certain theoretical and computational challenges. This thesis proposes a general penalized likelihood-based estimation approach for normal linear factor analysis models. The framework builds upon differentiable approximations of non-differentiable penalties and a theoretically founded definition of degrees of freedom. The employed optimization algorithm exploits second-order analytical derivative information and is integrated with an automatic tuning parameter selection procedure that finds the optimal value of the tuning without resorting to grid-searches. Some theoretical aspects of the penalized estimator are discussed. The proposed approach is evaluated in an extensive simulation study and illustrated using a psychometric data set. As a meaningful addition, the illustrated framework is extended to multiple-group factor analysis models, which are commonly used in cross-national surveys. The employed penalty simultaneously induces sparsity and cross-group equality of loadings and intercepts. The automatic procedure proves particularly useful in this challenging context, as it allows for the estimation of the multiple tuning parameters that compose the penalty term in a fast, stable and efficient way. The merits of the proposed technique are demonstrated through numerical and empirical examples. All the necessary routines are integrated into the R package GJRM to enhance reproducible research and transparent dissemination of results.