Structural Equation Modeling
(SEM) With Latent Variables
Steps In
Structural Equation Modeling
1. Model specification
2. Identification
3. Estimation
4. Testing fit
Measurement Model (1)
• Specifying the relationship between the latent variables and the observed variables
• Answers the questions:
1) To what extent are the observed variables actually measuring the hypothesized latent variables?
2) Which observed variable is the best measure of a particular latent variable?
Measurement Model (2)
• The relationships between the observed variables and the latent variables are described by factor loadings • Factor loadings provide information about the extent
to which a given observed variable is able to measure the latent variable. They serve as validity coefficients. • Measurement error is defined as that portion of an
observed variable that is measuring something other than what the latent variable is hypothesized to
Measurement Model (3)
• Measurement error could be the result of:
– An unobserved variable that is measuring some other latent variable
– Unreliability
Structural Model
• The researcher specifies the structural model to allow for certain relationships among the latent variables depicted by lines or arrows
• In the path diagram, we specified that Ability and Achievement were related in a specific way. That is, intelligence had some influence on later
Structural Model (2)
• The structural equation addresses the
following questions:
– Are Ability and Achievement related?
– Exactly how strong is the influence of Ability on Achievement?
Example of a Complete
Structual Equation Model
• We can specify a model to further duscuss how to diagram a model, specify the equations related to the model and discuss the “effects” apparent in the model. The example we use is a model of
educational achievement and aspirations.
• Figure 2 shows there are four latent variables (depicted by ellipses) two independent, home background (Home) and Ability and two
Example of a Complete
Structual Equation Model (2)
• Three of these latent variables are assessed
by two indicator variables and one latent
variable, home background, is assessed by
three indicator variables. The indicator
Covariance
• SEM involves the decomposition of
covariances
• There are different types of
covariance matrices:
1) Among the observed variables
Covariance (2)
• Types of covariance
1) Among the observed variables
2) Among the latent exogenous variables
Set the covariance between IQ and HOME to 0
IQ
HOME
Covariance (3)
3) Among the equation prediction errors
Set the error covariance between Legal and Profess free Religion
Experience
Legal Error
Profess Error
V1 F1
E1 E3
V2 F2
Total, Direct and Indirect Effects
• There is a direct effect between two latent variables when a single directed line or arrow connects them • There is an indirect effect between two variables
when the second latent variable is connected to the first latent variable through one or more other
latent variables
