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Products of Variables in Structural Equation Models.

Steven Boker1, Timo von Oertzen2, Joshua N Pritikin3

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Structural Equation Modeling : a Multidisciplinary Journal
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Summary
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A new Products of Variables Model (PoV) decomposes variable products in structural equation models (SEM). This method enables estimation of interactions and moderators, advancing SEM capabilities.

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Area of Science:

  • Statistics
  • Psychometrics
  • Quantitative Psychology

Background:

  • Structural Equation Models (SEM) are widely used for analyzing complex relationships between variables.
  • Existing SEM frameworks often struggle to directly model the variance and means of product terms, limiting the analysis of interactions and higher-order effects.

Purpose of the Study:

  • To introduce a general method for decomposing variables that are products of other variables within SEM.
  • To establish a new category of SEM, termed Products of Variables Models (PoV), to handle such decompositions.
  • To demonstrate the utility of PoV models for estimating interactions, latent variable moderators, and squared terms.

Main Methods:

  • Analytically derived expected means and covariances for simple products of two variables.
  • Algebraic investigation into the identifiability of PoV models based on multiplicand centering.
  • Implementation of the PoV method in statistical software (OpenMx and Ωnyx).

Main Results:

  • The PoV method successfully decomposes product variables into their sources of variance.
  • PoV models allow for the estimation of interactions between latent variables, latent variable moderators, and manifest moderators (even with missing data).
  • Identifiability of PoV models is achieved when multiplicands have non-zero means, while centered multiplicands lead to unidentified models.

Conclusions:

  • The Products of Variables Model (PoV) offers a significant advancement in structural equation modeling.
  • This new SEM category provides practical solutions for modeling complex interactions and higher-order terms.
  • The successful implementation and simulation studies validate the robustness and applicability of the PoV method.