01a SOC 681 Structual Equation Modeling with Latent Variables
Structural Equation Modeling
(SEM) With Latent Variables
James G. Anderson, Ph.D.
Purdue University
Steps In
Structural Equation Modeling
1.
2.
3.
4.
5.
Model specification
Identification
Estimation
Testing fit
Respecification
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?
3) To what extent are the observed variables actually
measuring something other than the hypothesized
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
measure. It serves as a measure of reliability.
Measurement Model (3)
• Measurement error could be the result of:
– An unobserved variable that is measuring some
other latent variable
– Unreliability
– A secondorder factor
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
achievement. Thus, one result from the structural
model is an indication of the extent to which these
a priori hypothesized relationships are supported
by our sample data.
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?
– Could there be other latent variables that we
need to consider to get a better understanding
of the influence 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
dependent, aspirations (Aspire) and achievement
(Achieve).
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
variables are depicted in rectangles.
Covariance
• SEM involves the decomposition of
covariances
• There are different types of
covariance matrices:
1)
2)
3)
4)
Among the observed variables
Among the latent exogenous variables.
Among the equation prediction errors
Among the measurement errors
Covariance (2)
•
Types of covariance
1) Among the observed variables
2) Among the latent exogenous variables
IQ
ACH
HOME
Set the covariance between IQ and HOME to 0
Covariance (3)
3) Among the equation prediction errors
Religion
E1
Legal
V1
E3
E2
F1
Profess
Experience
V2
E4
Error
Error
F2
Set the error covariance between Legal and Profess free
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
• The total effect between two latent variables is the
sum of any direct effect and all indirect effects that
connect them.
(SEM) With Latent Variables
James G. Anderson, Ph.D.
Purdue University
Steps In
Structural Equation Modeling
1.
2.
3.
4.
5.
Model specification
Identification
Estimation
Testing fit
Respecification
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?
3) To what extent are the observed variables actually
measuring something other than the hypothesized
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
measure. It serves as a measure of reliability.
Measurement Model (3)
• Measurement error could be the result of:
– An unobserved variable that is measuring some
other latent variable
– Unreliability
– A secondorder factor
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
achievement. Thus, one result from the structural
model is an indication of the extent to which these
a priori hypothesized relationships are supported
by our sample data.
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?
– Could there be other latent variables that we
need to consider to get a better understanding
of the influence 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
dependent, aspirations (Aspire) and achievement
(Achieve).
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
variables are depicted in rectangles.
Covariance
• SEM involves the decomposition of
covariances
• There are different types of
covariance matrices:
1)
2)
3)
4)
Among the observed variables
Among the latent exogenous variables.
Among the equation prediction errors
Among the measurement errors
Covariance (2)
•
Types of covariance
1) Among the observed variables
2) Among the latent exogenous variables
IQ
ACH
HOME
Set the covariance between IQ and HOME to 0
Covariance (3)
3) Among the equation prediction errors
Religion
E1
Legal
V1
E3
E2
F1
Profess
Experience
V2
E4
Error
Error
F2
Set the error covariance between Legal and Profess free
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
• The total effect between two latent variables is the
sum of any direct effect and all indirect effects that
connect them.