Lecture Notes: Structural Equation Modelling (SEM)

Introduction to SEM

• SEM Definition: Structural Equation Modelling (SEM) is not a single technique but a general modelling framework integrating different multivariate techniques.
• Disciplines Involved:
  • Measurement theory from psychology
  • Factor analysis from psychology
  • Statistics
  • Path analysis from epidemiology and biology
  • Regression modelling from statistics
  • Simultaneous equations from econometrics
• Dynamic Nature: SEM is a complex and evolving environment, integrating new modelling techniques over time.

Research Questions Suitable for SEM

• Complex and Multifaceted Constructs: Particularly useful for psychological and social psychological concepts, which are difficult to measure and often include measurement errors.
• Systems of Relationships: Suitable for models with numerous dependent variables affecting each other in complex systems, ideal for causal system modelling.
• Indirect or Mediated Effects: Useful for analyzing indirect effects where a variable influences another through a mediator.

Terminology and Names

• Covariance Structure Analysis: SEM analyzes covariance matrices.
• Analysis of Moment Structures: SEMs analyze both covariances and means.
• LISREL Model: Named after a popular software for fitting SEMs.
• Causal Modelling: A controversial term, as causal claims depend on research design, not just statistical models.

Software for SEM

• Popular Software: LISREL, M+, EQS, AMOS, R, Stata.
• Software Selection: No single recommendation; each has its pros and cons.

SEM as Path Analysis Using Latent Variables

• Latent Variables: Concepts not directly observable (e.g., intelligence, social capital, trust).
• Observable Indicators: Measured variables believed to be caused by latent constructs.

Measurement Error and True Score

• True Score Equation: X = T + E, where X is the observed indicator, T is the true score, and E is the error.
• Systematic vs Random Error: Systematic error causes bias, while random error averages to zero.

Identifying Latent Variables

• Multiple Indicators: Required to over-identify the true score equation and estimate T and E.
• Latent Variable Models:
  • Principal components analysis
  • Factor analysis
  • Latent class models
• Common Factor Model: Uses multiple indicators for more accurate latent variable measurement.

Benefits of Latent Variable Modelling

• Complex Constructs: Helps measure multifaceted constructs like happiness with multiple indicators.
• Error Reduction: Removes or reduces random error, improving precision and reducing bias.

Path Analysis

• Diagrammatic Representation: Visual representation of models rather than equations.
• Focus on Direct and Indirect Effects: Important for studying complex relationships.

Standardized Notation in Path Analysis

• Measured Latent Variable: Represented as an ellipse.
• Observed Variable: Represented as a rectangle.
• Error Variance: Represented as a small circle.
• Covariance Path: Curved double-headed arrow for non-directional associations.
• Directional Path: Single-headed straight arrow indicating causality.

Examples of Path Diagrams

• Simple and Multiple Regression Models: Bivariate and multiple linear regression models.
• Indirect Effects: Models illustrating indirect effects through mediators.

Conclusion

• Combining Latent Variables with Path Analysis: Represents structural equation models when path diagrams include latent variables.

These notes encapsulate the key points discussed in the lecture on Structural Equation Modelling (SEM), providing a foundational understanding of the techniques, applications, and theoretical underpinnings of SEM.