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Abstract: Comparing Semiparametric and Parametric Methods for Modeling Interactions Among Latent Variables.

Ruth E Baldasaro1, Daniel J Bauer1

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Area of Science:

  • Statistics
  • Psychometrics
  • Econometrics

Background:

  • Estimating interactions among latent variables is crucial in many fields.
  • Existing parametric methods often assume specific functional forms, which may not reflect reality.
  • A need exists for flexible, exploratory methods to model unknown interaction forms.

Purpose of the Study:

  • To evaluate a semiparametric approach (SEMM) for estimating latent variable interactions of unknown functional form.
  • To compare the performance of SEMM against two parametric approaches: latent moderated structures and unconstrained product-indicator.
  • To assess the bias and accuracy of these methods under different data-generating functional forms.

Main Methods:

  • A simulation study was conducted using data generated from four functional forms: main effects only, quadratic trend, bilinear interaction, and exponential interaction.
  • Structural Equation Mixture Models (SEMM) were fit to the data to approximate interactions.
  • Performance was assessed by comparing model-implied surfaces to true data-generating surfaces using bias and root mean squared error of approximation.

Main Results:

  • Parametric approaches were more efficient but could yield biased estimates when the assumed functional form did not match the data-generating form (e.g., quadratic or exponential interactions).
  • The SEMM approach demonstrated a low level of bias across various nonlinear surfaces, approximating the true data-generating surface effectively.
  • For main effects and bilinear interactions, SEMM showed comparable or slightly higher bias than parametric methods, but excelled in flexibility.

Conclusions:

  • The SEMM approach offers a valuable, relatively unbiased method for approximating diverse nonlinear relationships among latent variables when the functional form is unknown.
  • Parametric approaches are efficient but require correct specification of the interaction's functional form to avoid bias.
  • SEMM provides a more robust and exploratory alternative for interaction estimation in latent variable modeling.