Structural Equation Modeling SEM With Latent Variables

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Structural Equation Modeling (SEM) With Latent
Variables

• James G. Anderson, Ph.D.
• Purdue University

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Steps In Structural Equation Modeling

• Model specification
• Identification
• Estimation
• Testing fit
• Respecification

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Measurement Model (1)

• Specifying the relationship between the latent
  variables and the observed variables
• Answers the questions
• To what extent are the observed variables
  actually measuring the hypothesized latent
  variables?
• Which observed variable is the best measure of a
  particular latent variable?
• To what extent are the observed variables
  actually measuring something other than the
  hypothesized latent variable?

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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.

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Measurement Model (3)

• Measurement error could be the result of
• An unobserved variable that is measuring some
  other latent variable
• Unreliability
• A second-order factor

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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.

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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?

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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).

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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.

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Covariance

• SEM involves the decomposition of covariances
• There are different types of covariance matrices
• Among the observed variables
• Among the latent exogenous variables.
• Among the equation prediction errors
• Among the measurement errors

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Covariance (2)

• Types of covariance
• Among the observed variables
• Among the latent exogenous variables
• Set the covariance between IQ and HOME to 0

IQ
ACH
HOME
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Covariance (3)

• Among the equation prediction errors
• Set the error covariance between Legal and
  Profess free

Religion
Legal
Error
V1
F1
E1
E3
Profess
Error
Experience
V2
F2
E2
E4
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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.