Penalized Least Squares for Structural Equation Modeling with Ordinal Responses.

Area of Science:

• Statistics
• Machine Learning
• Statistical Modeling

Background:

• Sparse statistical modeling is a key research area in statistics and machine learning.
• Identifying the true sparsity pattern in models is challenging and often requires specialized estimation procedures like the least absolute shrinkage and selection operator (lasso).

Purpose of the Study:

• To develop a penalized least squares (PLS) method for structural equation modeling (SEM) specifically designed for ordinal data.
• To compare the performance of the proposed PLS method against existing penalized likelihood (PL) methods.

Main Methods:

• A penalized least squares (PLS) approach was developed using an underlying response model and a least squares fitting function.
• Numerical simulations were conducted to evaluate PLS against penalized likelihood (PL) based on mean square error, absolute bias, and model correctness.
• A hybrid PLS method was proposed, combining PL for optimal sparsity pattern selection and unpenalized least squares (LS) for parameter estimation.
• The PLS method was extended to handle mixed-type data and multi-group analysis.

Main Results:

• The penalized least squares (PLS) method demonstrated effectiveness in structural equation modeling for ordinal data.
• Simulations indicated that PLS is a competitive alternative to penalized likelihood (PL) methods.
• The proposed hybrid PLS method showed potential for improving upon both PL and PLS.
• Extensions of PLS for mixed-type and multi-group analyses were successfully developed.

Conclusions:

• The developed penalized least squares (PLS) method offers a robust approach for sparse structural equation modeling with ordinal data.
• The hybrid PLS strategy provides an effective way to select sparsity patterns and estimate parameters.
• The R package lslx facilitates the implementation of these advanced statistical modeling techniques.