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

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

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Data Preparation

• Estimation of missing data
• Creation of scales and indices
• Descriptive statistics to include
• Examination for outliers
• skewness and kurtosis
• Transformation of variables

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Measurement Model (1)

• Specifying the relationship between the latent
  variables and the observed variables
• Answers these 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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Setting the Error Variance

• Error variance can be set to 0 if you have a
  single indicator of the latent variable and no
  information about its reliability
• Error Variance (1-Reliability) Variance of the
  Observed Score if you know the reliability of the
  indicator

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Creating a Latent Variable from Multiple
Indicators

• Exploratory factor analysis can be used with
  multiple indicators of a construct to determine
  the number of factors and which indicators are
  associated with each factor.
• Confirmatory factor analysis can then be used to
  test the fit of the measurement model.

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Example of a Complete Structural Equation Model

• We can specify a model to further discuss 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 3 shows there are four latent variables
  (depicted by ellipses), three exogenous variables
  (knowledge, Value and Satisfaction) and one
  endogenous (performance).

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Variables

• Performance Planning, Organization,
  controlling, coordinating and directing a farm
  cooperative
• Knowledge Knowledge of economic phases of
  management directed toward profit-making
• Value-Tendency to rationally evaluate means to an
  economic end
• Satisfaction - Gratification from performing the
  managerial role

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Structural Model (1)

• 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
  Performance is related to Knowledge, Value and
  Satisfaction in a specific way. 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
• Is Performance related to the three predictor
  variables?
• Exactly how strong is the influence of each
  variable on Performance?
• How well does the model fit the data?

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Example of a Complete Structural Equation Model
(2)

• Each of the four latent variables is assessed by
  two indicator variables. The indicator variables
  are depicted in rectangles.

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Model Building

• If the original model does not provide an
  acceptable fit to the data, alternative models
  can be tested.
• The standardized residuals and modification
  indices can be used to determine how to modify
  the model to achieve a better fit to the data.

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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

IQ
ACH
HOME
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Covariance (3)

• Among the equation prediction errors

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.