Title: Structural Equation Modeling: An Overview
1Structural Equation Modeling An Overview
- Pamela Paxton
- Ohio State University
2What are Structural Equation Models?
- Also known as
- Covariance structure models
- Latent variable models
- LISREL models
- Structural Equations with Latent Variables
3What are Structural Equation Models?
- Special cases
- ANOVA
- Multiple regression
- Path analysis
- Confirmatory Factor Analysis
- Recursive and Nonrecursive systems
4What 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
5What 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
6What 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
7What 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
8What 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)
9Notation
?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
10Notation
- ? Latent Endogenous Variable
- ? Latent Exogenous Variable
- ? Unexplained Error in Model
- x y Observed Variables
- d e Measurement Errors
- ?, ß, ? Coefficients
11Notation
- Two components to a SEM
- Latent variable model
- Relationship between the latent variables
- Measurement model
- Relationship between the latent and observed
variables
12Notation
- Covariance Matrixes of Interest
- F
- ?
- Td
- Te
13Example 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
14Latent Variables
- Variables of Interest
- Not directly measured
- Common
- Intelligence
- Trust
- Democracy
- Diseases
- Disturbance variables
15Three Types of SEM
- Classic Econometric
- Multiple equations
- One indicator per latent variable
- No measurement error
16Classic Econometric
Citations y3
ß43
ß32
?31
Publications y2
ß42
ß31
Quality rating y4
ß41
Size of dept. y1
?11
?41
Private x1
17Classic Econometric
Ethnic homogeneity
Noncore position
industrialization 1980
democracy 1982
democracy 1991
trust 1980
trust 1990
associations 1980
associations 1990
18Recursive / 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
19Recursive
y2
x1
y3
y3
x1
y1
y2
x2
x3
20Nonrecursive
x2
y1
x1
y2
y1
x1
y2
x2
y3
x3
21Confirmatory 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
22Confirmatory 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
23General Model
- Includes latent variable model
- Relationship between the latent variables
- And measurement model
- Relationship between latent variables and
observed variables
24General 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
25General 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
26General 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
27Six Steps to Modeling
- Specification
- Implied Covariance Matrix
- Identification
- Estimation
- Model Fit
- Respecification
28Specification
- 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?
29Identification
- Are there unique values for parameters?
- Property of model, not data
- 10 x y
30Identification
- Underidentified
- Just identified
- Overidentified
31Identification
- 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
32Identification
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.
33Model Fit
- Component Fit
- Use Substantive Experience
- Are signs correct?
- Any nonsensical results?
- R2s for individual equations
- Negative error variances?
- Standard errors seem reasonable?
34Model 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
35Model 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
36Model Fit
?2 223, df5, p.000 IFI .87 RMSEA
.25
N801
37Respecification
- Theory!
- Dimensionality?
- Correct pattern of loadings?
- Correlated errors of measurement?
- Other paths?
- Modification Indexes
- Residuals
38Respecification
?2 3.8, df2, p.15 IFI 1.0 RMSEA
.03
N801
39Useful 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