# Structural Equation Modeling (SEM)

**Structural Equation Modeling (SEM)** is a statistical technique that allows for the testing and estimation of complex relationships among observed and latent variables. It is an extension of general linear models that encompasses factor analysis, path analysis, and simultaneous equation models, among other techniques. SEM allows researchers to test hypotheses about causal relationships among variables, often in the context of confirmatory data analysis.

## History

The development of SEM can be traced back to the work of Sewall Wright in the early 20th century, who introduced path analysis. Over the years, SEM has been enriched by the work of scholars like Karl Jöreskog, who developed the LISREL program, one of the first software packages designed to conduct SEM.

## Components

- Observed Variables: Also known as manifest variables, these are variables that are directly measured in a study.
- Latent Variables: These are variables that are not directly measured but are estimated in the model from observed variables, often serving to represent abstract concepts like intelligence, satisfaction, or motivation.

## Estimation Techniques

- Maximum Likelihood: This is the most commonly used method for parameter estimation in SEM. It estimates parameters that maximize the likelihood function given the observed data.
- Generalized Least Squares: This is another method for estimating parameters but is less commonly used than Maximum Likelihood.

## Model Fit

**Goodness of Fit**

Several statistics can be used to assess the goodness of fit of an SEM model, including the Chi-Square Test of Model Fit, Root Mean Square Error of Approximation (RMSEA), and Comparative Fit Index (CFI), among others.

## Model Comparisons

SEM allows for the comparison of nested models, which can be useful for hypothesis testing.

## Applications

- Psychology: Testing theories of cognitive processes or personality traits
- Marketing: Understanding consumer behavior and attitudes
- Medicine: Modeling the relationship between symptoms and diseases
- Sociology: Investigating social influence and structure

## Software

- LISREL
- AMOS
- Mplus
- R packages like lavaan and sem

## Limitations and Criticisms

- Complexity: SEM models can become very complex, making them hard to interpret.
- Identification Issues: The model must be identified mathematically, meaning there must be enough data to estimate the desired parameters.
- Large Sample Requirements: SEM generally requires large sample sizes for reliable estimates.