Structural Equation Modeling
Structural equation models impose a set of restrictions on the variance-covariance matrix
of observed variables.
For SEM at least two variables are needed. However, it really is useful when it is used for
complex systems with multiple variables, relationships and even unobservable constructs.
Covariance matrix: Is a mathematical representation showing how pairs of variables
(observed data points) vary together.
o This table lists the variance of individual variables on the diagonal
o Covariances between pairs of variables are shown in the oF-diagonal
elements.
SEM specifies how variables are expected to interact in this matrix, reducing its complexity.
• Variance: Indicates how much a single variable varies, how spread out the data for
a single variable is.
• Covariance: Shows how two variables vary together.
Saturated model: A saturated model in SEM is the most flexible model possible. It
assumes that all possible relationships between variables are included and estimates
them.
• Every observed variable is allowed to relate to every other variable (through
covariances)
• NO restrictions are imposed, so it fully fits the data (no degrees of freedom)
A saturated model always fits the data perfectly because it does not impose any structure
or constraints. à A perfect fit does not mean the model is useful, its too flexible and it does
not fit any theory.
A saturated model is used as a reference point to compare other, more restricted models. It
helps researchers understand the full covariance structure of the data before imposing
theoretical constraints.
, Restricted structural equation model: in a restricted structural equation model, certain
theoretical relationships are specified between variables.
Here X1 influences X2, and X2 influences X3.
• Variances and covariances are derived based on these relationships
The adequacy of a restricted model is tested by comparing the observed covariance matrix
with the covariance matrix implied by these equations.
SEM can be used for:
• Confirmatory Research: Testing theories
• Explanatory research: Explaining dependent variables
• Explanatory research: Recognizing patterns in data
• Predictive research: Determining values for new individual cases
• Descriptive research: Reporting variable levels
Auxiliary theory: in structural equation modeling is a supporting framework that helps
bridge the gap between abstract concepts and the measurable variables used to represent
them. It explains how we can operationalize or measure these concepts, which are not
directly observable.
The two main auxiliary theories are:
1. Measurement theory: assumes that a latent variable (an abstract construct, which
cannot be measured directly) is the underlying cause of a group of observable
variables.
§ Observed variables are seen as the eFect of the latent variable.
§ The observed variables are independent of each other after accounting for
the latent variable. The latent variable thus explains the shared variance
among observed variables, making them unrelated to one another directly.
Measurement theory starts with a theoretical concept, operationalizes it as a latent
variable, and connects it to observable data through proxy measures.
Structural equation models impose a set of restrictions on the variance-covariance matrix
of observed variables.
For SEM at least two variables are needed. However, it really is useful when it is used for
complex systems with multiple variables, relationships and even unobservable constructs.
Covariance matrix: Is a mathematical representation showing how pairs of variables
(observed data points) vary together.
o This table lists the variance of individual variables on the diagonal
o Covariances between pairs of variables are shown in the oF-diagonal
elements.
SEM specifies how variables are expected to interact in this matrix, reducing its complexity.
• Variance: Indicates how much a single variable varies, how spread out the data for
a single variable is.
• Covariance: Shows how two variables vary together.
Saturated model: A saturated model in SEM is the most flexible model possible. It
assumes that all possible relationships between variables are included and estimates
them.
• Every observed variable is allowed to relate to every other variable (through
covariances)
• NO restrictions are imposed, so it fully fits the data (no degrees of freedom)
A saturated model always fits the data perfectly because it does not impose any structure
or constraints. à A perfect fit does not mean the model is useful, its too flexible and it does
not fit any theory.
A saturated model is used as a reference point to compare other, more restricted models. It
helps researchers understand the full covariance structure of the data before imposing
theoretical constraints.
, Restricted structural equation model: in a restricted structural equation model, certain
theoretical relationships are specified between variables.
Here X1 influences X2, and X2 influences X3.
• Variances and covariances are derived based on these relationships
The adequacy of a restricted model is tested by comparing the observed covariance matrix
with the covariance matrix implied by these equations.
SEM can be used for:
• Confirmatory Research: Testing theories
• Explanatory research: Explaining dependent variables
• Explanatory research: Recognizing patterns in data
• Predictive research: Determining values for new individual cases
• Descriptive research: Reporting variable levels
Auxiliary theory: in structural equation modeling is a supporting framework that helps
bridge the gap between abstract concepts and the measurable variables used to represent
them. It explains how we can operationalize or measure these concepts, which are not
directly observable.
The two main auxiliary theories are:
1. Measurement theory: assumes that a latent variable (an abstract construct, which
cannot be measured directly) is the underlying cause of a group of observable
variables.
§ Observed variables are seen as the eFect of the latent variable.
§ The observed variables are independent of each other after accounting for
the latent variable. The latent variable thus explains the shared variance
among observed variables, making them unrelated to one another directly.
Measurement theory starts with a theoretical concept, operationalizes it as a latent
variable, and connects it to observable data through proxy measures.
