Structural Equation Modeling: An Overview

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Structural Equation Modeling An Overview

• Pamela Paxton
• Ohio State University

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What are Structural Equation Models?

• Also known as
• Covariance structure models
• Latent variable models
• LISREL models
• Structural Equations with Latent Variables

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What are Structural Equation Models?

• Special cases
• ANOVA
• Multiple regression
• Path analysis
• Confirmatory Factor Analysis
• Recursive and Nonrecursive systems

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What are Structural Equation Models?

• SEM associated with path diagrams

intelligence
test 2
test 3
test 4
test 5
test 1
d1
d2
d4
d3
d5
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What are Structural Equation Models?
Latent variables, factors, constructs
Observed variables, measures, indicators,
manifest variables
Direction of influence, relationship from one
variable to another
Association not explained within the model
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What are Structural Equation Models?
e1
e 2
e 3
Depress 1
Depress 2
Depress 3
Family support
?1
depression
?2
Physical health
Self rated closeness
Spousal rating
Kids rating
d1
d2
d3
Self rating
MD rating
visits to MD
e4
e 5
e 6
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What are Structural Equation Models?

• What can you do with these models?
• Latent and Observed Variables
• Multiple indicators of same concept
• Measurement error
• Restrictions on model parameters
• Tests of model fit

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What are Structural Equation Models?

• What cant you do?
• Prove causation
• Prove a model is correct

All models
Models consistent with data
Models consistent with reality
(Mueller 1997)
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Notation
?4
?5
?6
?7
?8
?9
?10
?11
?1
?2
ß21
?2
?1
?11
?21
?1
?1 industrialization ?1 democracy time 1 ?2
democracy time 2 x1-x3 indus. indicators, e.g.,
energy y1-y4 democ. indicators time 1 y5-y8
democ. indicators time 2
?1
?2
?3
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Notation

• ? Latent Endogenous Variable
• ? Latent Exogenous Variable
• ? Unexplained Error in Model
• x y Observed Variables
• d e Measurement Errors
• ?, ß, ? Coefficients

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Notation

• Two components to a SEM
• Latent variable model
• Relationship between the latent variables

• Measurement model
• Relationship between the latent and observed
  variables

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Notation

• Covariance Matrixes of Interest
• F
• ?
• Td
• Te

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Example Trust in Individuals
Trust in Individuals
?1
?11
1
?21
people are helpful (x1)
people can be trusted (x2)
people are Fair (x3)
d1
d2
d3
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Latent Variables

• Variables of Interest
• Not directly measured
• Common
• Intelligence
• Trust
• Democracy
• Diseases
• Disturbance variables

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Three Types of SEM

• Classic Econometric
• Multiple equations

• One indicator per latent variable
• No measurement error

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Classic Econometric
Citations y3
ß43
ß32
?31
Publications y2
ß42
ß31
Quality rating y4
ß41
Size of dept. y1
?11
?41
Private x1
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Classic Econometric
Ethnic homogeneity
Noncore position
industrialization 1980
democracy 1982
democracy 1991
trust 1980
trust 1990
associations 1980
associations 1990
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Recursive / Nonrecursive

• Recursive
• Direction of influence one direction
• No reciprocal causation
• No feedback loops
• Disturbances not correlated
• Nonrecursive
• Either reciprocal causation, feedback loops, or
  correlated disturbances

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Recursive
y2
x1
y3
y3
x1
y1
y2
x2
x3
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Nonrecursive
x2
y1
x1
y2
y1
x1
y2
x2
y3
x3
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Confirmatory Factor Analysis

• Latent variables
• Measurement error
• No causal relationship between latent variables

x vector of observed indicators ?x matrix of
factor loadings ? vector of latent variables d
vector of measurement errors
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Confirmatory Factor Analysis
Trust in Individuals
?1
?11
1
?21
people are helpful (x1)
people can be trusted (x2)
people are Fair (x3)
d1
d2
d3
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General Model

• Includes latent variable model
• Relationship between the latent variables
• And measurement model
• Relationship between latent variables and
  observed variables

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General Model

• Latent Variable Model

? vector of latent endogenous variables ?
vector of latent exogenous variables ? vector
of disturbances ? coefficient matrix for ? on ?
effects G coefficient matrix for ? on ? effects
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General Model

• Measurement Model

x indicators of ? ?x factor loadings of ? on
x y indicators of ? ?y factor loadings of ?
on y d measurement error for x e measurement
error for y
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General SEM
?4
?5
?6
?7
?8
?9
?10
?11
?1
?2
ß21
?2
?1
?11
?21
?1
?1 industrialization ?1 democracy time 1 ?2
democracy time 2 x1-x3 indus. indicator, e.g.,
energy y1-y4 democ. indicators time 1 y5-y8
democ. indicators time 2
?1
?2
?3
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Six Steps to Modeling

• Specification
• Implied Covariance Matrix
• Identification
• Estimation
• Model Fit
• Respecification

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Specification

• Theorize your model
• What observed variables?
• How many observed variables?
• What latent variables?
• How many latent variables?
• Relationship between latent variables?
• Relationship between latent variables and
  observed variables?
• Correlated errors of measurement?

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Identification

• Are there unique values for parameters?
• Property of model, not data
• 10 x y

• 2, 8
• -1, 11
• 4, 6

• x y

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Identification

• Underidentified
• Just identified
• Overidentified

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Identification

• Rules for Identification
• By type of model
• Classic econometric
• e.g., recursive rule
• Confirmatory factor analysis
• e.g., three indicator rule
• General Model
• e.g., two-step rule

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Identification
Trust in Individuals
?1
?11
1
?21
people are helpful (x1)
people can be trusted (x2)
people are Fair (x3)
d1
d2
d3

• Identified? Yes, by 3-indicator rule.

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Model Fit

• Component Fit
• Use Substantive Experience
• Are signs correct?
• Any nonsensical results?
• R2s for individual equations
• Negative error variances?
• Standard errors seem reasonable?

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Model Fit

• How well does our model fit the data?
• The Test Statistic (?2)
• T(N-1)F
• df½(pq)(pq1) - of parameters
• p number of ys
• q number of xs
• SS(?)
• Statistical power

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Model Fit

• Many goodness-of-fit statistics
• Tb chi-square test statistic for baseline model
• Tm chi-square test statistic for hypothesized
  model
• dfb degrees of freedom for baseline model
• dfm degrees of freedom for hypothesized model

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Model Fit
?2 223, df5, p.000 IFI .87 RMSEA
.25
N801
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Respecification

• Theory!
• Dimensionality?
• Correct pattern of loadings?
• Correlated errors of measurement?
• Other paths?
• Modification Indexes
• Residuals

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Respecification
?2 3.8, df2, p.15 IFI 1.0 RMSEA
.03
N801
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Useful References

• Book from which this talk is drawn Bollen,
  Kenneth A. 1989. Structural Equations with
  Latent Variables. New York Wiley.
• Ed Rigdons website www.gsu.edu/mkteer/
• Archives of SEMNET listserv bama.ua.edu/archives/
  semnet.html