Penalized Least Squares for Structural Equation Modeling with Ordinal Responses

Abstract

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.

Keywords:

• Structural equation modeling
• factor analysis
• lasso
• penalized least squares
• polychoric correlation

Article information

Conflict of interest disclosures: The author signed a form for disclosure of potential conflicts of interest. The author did not report any financial or other conflicts of interest in relation to the work described.

Ethical principles: The author affirms having followed professional ethical guidelines in preparing this work. These guidelines include obtaining informed consent from human participants, maintaining ethical treatment and respect for the rights of human or animal participants, and ensuring the privacy of participants and their data, such as ensuring that individual participants cannot be identified in reported results or from publicly available original or archival data.

Funding: This work was supported by Grant MOST107-2410-H-006-057-MY2 from the Ministry of Science and Technology in Taiwan.

Role of the funders/sponsors: None of the funders or sponsors of this research had any role in the design and conduct of the study; collection, management, analysis, and interpretation of data; preparation, review, or approval of the manuscript; or decision to submit the manuscript for publication.

Acknowledgements: The ideas and opinions expressed herein are those of the author alone, and endorsement by the author’s institution is not intended and should not be inferred.

Notes

1 A baseline model is required to calculate CFI and NNFI. In lslx, the baseline model includes a saturated threshold structure. We found that such implementation can result in unreasonably large values for CFI and NNFI. Therefore, CFI and NNFI, as presented here, were calculated by only considering the misfit of the correlation structures. The baseline model here assumed that the correlation matrix is an identity matrix.

Additional information

Funding

This work was supported by Grant MOST107-2410-H-006-057-MY2 from the Ministry of Science and Technology in Taiwan.