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A Note on the Connection Between Trek Rules and Separable Nonlinear Least Squares in Linear Structural Equation

Maximilian S Ernst1,2, Aaron Peikert3,4,5, Andreas M Brandmaier3,5,6

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Summary
This summary is machine-generated.

Separable nonlinear least squares (SNLLS) estimation now applies to all linear structural equation models (SEMs). This method improves convergence and computation time for SEMs, offering analytical solutions for specific model classes.

Keywords:
Gaussian graphical modelRAM notationgraph theoryleast squares estimationnumerical optimization

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Area of Science:

  • Statistics
  • Econometrics
  • Psychometrics

Background:

  • Separable nonlinear least squares (SNLLS) is an estimation technique known for improving convergence and computation time in models like neural networks.
  • SNLLS is applicable to models where a subset of parameters linearly influences the objective function.
  • Previous work demonstrated SNLLS applicability to factor analysis models.

Purpose of the Study:

  • To generalize the applicability of separable nonlinear least squares (SNLLS) estimation to all linear structural equation models (SEMs) specified in RAM notation.
  • To demonstrate that undirected effects (variances, covariances) and mean parameters enter the objective linearly in SEMs.
  • To provide an efficient gradient expression for SNLLS in SEMs.

Main Methods:

  • Generalizing SNLLS applicability to all linear SEMs using RAM notation.
  • Employing trek rules to connect graphical SEM representations with covariance parametrization.
  • Deriving an efficient gradient expression for SNLLS in SEMs.

Main Results:

  • SNLLS estimation is applicable to all linear structural equation models (SEMs).
  • Undirected effects and mean parameters enter the objective linearly, requiring only directed effects to be estimated iteratively.
  • For SEMs without unknown directed effects, SNLLS allows for analytical computation of least squares estimates.
  • Simulation results show SNLLS improves convergence rates and reduces iteration counts.

Conclusions:

  • SNLLS provides a more efficient estimation method for a broad class of linear structural equation models.
  • The findings facilitate faster and more reliable estimation of SEMs, particularly those with specific parameter structures.
  • This generalization enhances the practical utility of SNLLS in statistical and econometric modeling.