Penalized Least Squares for Structural Equation Modeling with Ordinal Responses

Statistical modeling with sparsity has become an active research topic in the fields of statistics and machine learning. Because the true sparsity pattern of a model is generally unknown aforehand, it is often explored by a sparse estimation procedure, like least absolute shrinkage and selection operator (lasso). In this study, a penalized least squares (PLS) method for structural equation modeling (SEM) with ordinal data is developed. PLS describes data generation by an underlying response approach, and uses a least squares (LS) fitting function to construct a penalized estimation criterion. A numerical simulation was used to compare PLS with existing penalized likelihood (PL) in terms of averaged mean square error, absolute bias, and the correctness of the model. Based on these empirical findings, a hybrid PLS was also proposed to improve both PL and PLS. The hybrid PLS first chooses an optimal sparsity pattern by PL, then estimates model parameters by an unpenalized LS under the model selected by PL. We also extended PLS to cases of mixed type data and multi-group analysis. All proposed methods could be realized in the R package lslx.