Exact Exploratory Bi-factor Analysis: A Constraint-Based Optimization Approach

Area of Science:

• Psychometrics
• Statistical Modeling

Background:

• Bi-factor analysis, a type of confirmatory factor analysis, is crucial in psychological and educational measurement.
• Specifying an explicit bi-factor structure is necessary for bi-factor model application.
• The bi-factor structure is often unknown, necessitating exploratory bi-factor analysis methods.

Purpose of the Study:

• To introduce a novel constraint-based optimization method for exploratory bi-factor analysis.
• To address the limitations of existing rotation-based methods that do not yield exact bi-factor loading structures.
• To provide a method for learning an exact bi-factor loading structure directly from data.

Main Methods:

• Formulating exploratory bi-factor analysis as a constrained optimization problem in a continuous domain.
• Utilizing a mathematical characterization of the bi-factor loading structure as equality constraints.
• Solving the optimization problem using an augmented Lagrangian method.

Main Results:

• The proposed constraint-based method successfully learns an exact bi-factor loading structure.
• This method overcomes the imprecision issues associated with previous rotation-based approaches.
• Simulation studies and a real data example demonstrate the method's effectiveness.

Conclusions:

• The developed constraint-based optimization method offers a powerful tool for exploratory bi-factor analysis.
• It provides an accurate way to determine bi-factor structures when they are not explicitly known.
• This advancement has significant implications for psychological and educational measurement research.