Structural Equation Modeling

Structural equation modeling (SEM) is a modeling framework that makes use of multiple prediction equations. It is also also known as covariance structure analysis, analysis of moment structures, or analysis of linear structural relationships.

Description

SEM is used for measurement error adjustment.

The first component is the measurement model, which is essentially a CFA. Terminology is also common between the two, e.g. factors, factor loadings, indicators, and so on. The most important distinction is that a causal direction is assumed; note the arrows below:

This is equivalent to a formulation like:

• X = α₁ + β₁x₁ + e₁

• X = α₂ + β₂x₂ + e₂

• X = α₃ + β₃x₃ + e₃

The second component is the structural model which specifies the mediation relationship of the latent factors. If a factor is predicted by other variables in the system, it is endogenous; otherwise it is exogenous.

The simplest formulation of a structural model might be Y = α_Y + β_YX + e_Y, but...

• there can be multiple predictive factors, e.g. Y ~ X + Z

• the outcome can itself have a measurement model, e.g. Y ~= y1 + y2 + y3

Most model estimation strategies require assuming:

1. a uniform variance for the outcome construct's residual, i.e. e_Y.

2. a uniform variance for the latent constructs, e.g. X.

See https://www.youtube.com/watch?v=NOWdrfQVWAI&t=2045s for an demonstration of why assumptions are required, and how many are needed.

All variables are assumed to be jointly normal. Failures of this assumption are sometimes addressed through deleting outlier observations or transforming variables.

Reading Notes

• Asymptotically distribution-free methods for the analysis of covariance structures, M. W. Browne, 1984

• A comparison of some methodologies for the factor analysis of non‐normal Likert variables; Bengt O. Muthén and David E. Kaplan; 1985

• Latent Variable Modeling in Heterogeneous Populations, Bengt O. Muthén, 1989

• Multilevel Covariance Structure Analysis, Bengt O. Muthén, 1994

• Corrections to Test Statistics and Standard Errors in Covariance Structure Analysis, Albert Satorra and Peter M. Bentler, 1994

• Structural equation modeling with robust covariances, Ke-Hai Yuan and Peter M. Bentler, 1998

• A Comparative Review of Interaction and Nonlinear Modeling; Edward E. Rigdon, Randall E. Schumacker, and Werner Wothke; 1998

• Robust transformation with applications to structural equation modelling; Ke-Hai Yuan, Wai Chan, and Peter M. Bentler; 2000

• Non-normality of Data in Structural Equation Models; Shengyi Gao, Patricia L. Mokhtarian, and Robert A. Johnston; 2008

• General Random Effect Latent Variable Modeling: Random Subjects, Items, Contexts, and Parameters, Tihomir Asparouhov and Bengt Muthén, 2014

• A closer look at random and fixed effects panel regression in structural equation modeling using lavaan, Henrik Kenneth Andersen, 2021

CategoryRicottone