Structural Equation Modeling: An Overview

Structural Equation Modeling: An Overview

P. Paxton

What are Structural Equation Models?

• Also known as:– Covariance structure models– Latent variable models– “LISREL” models– Structural Equations with Latent Variables


Special cases:ANOVAMultiple regressionPath analysisConfirmatory Factor AnalysisRecursive and Nonrecursive systems


• SEM associated with path diagrams

intelligence

test 1 test 2 test 3 test 4 test 5

δ1 δ2 δ3 δ4 δ5


Latent variables, factors, constructs

Observed variables, measures, indicators, manifest variables

Direction of influence, relationship from one variableto another

Association not explained within the model


Depress 1 Depress 2 Depress 3

Self rating MD rating # visits to MD

Self ratedcloseness

Spousalrating

Kids rating

Family support depression

Physical health

δ1 δ2 δ3

ε4 ε 5 ε 6

ε1 ε 2 ε 3

ζ1

ζ2


• 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


• What can’t you do?– Prove causation– Prove a model is “correct”

All models

Models consistent with data

Models consistent with reality

(Mueller 1997)

Notation

ε1

y1

ε2

y2

ε3

y3

ε4

y4

ε5

y5

ε6

y6

ε7

y7

ε8

y8

δ1

x1

δ2

x2

δ3

x3

η1

ξ1

η2

ζ1 ζ2β21

γ21

γ11

λ1 λ2 λ3

λ4 λ5 λ6 λ7 λ8 λ9 λ10 λ11

ξ1= industrializationη1 = democracy time 1η2 = democracy time 2x1-x3 = indus. indicators, e.g., energyy1-y4 = democ. indicators time 1y5-y8 = democ. indicators time 2

Notation

• η Latent Endogenous Variable• ξ Latent Exogenous Variable• ζ Unexplained Error in Model

• x & y Observed Variables• δ & ε Measurement Errors• λ, β, & γ Coefficients

Notation• Two components to a SEM

– Latent variable model• Relationship between the latent variables

ζΓξΒηη

δξΛx x εηΛy y

Measurement model• Relationship between the latent and observed variables

Notation

• Covariance Matrixes of Interest:– Φ– Ψ– Θδ

– Θε

Example: Trust in Individuals

δξΛx x Trust in Individuals

people arehelpful

(x1)

people canbe trusted

(x2)

people areFair (x3)

1

ξ1

δ1 δ2 δ3

11111 x

21212 x

313 x

)( 111 VAR

)(00

)(0

)(

3

2

1

VAR

VAR

VAR

λ11 λ21

Latent Variables

• Variables of Interest• Not directly measured• Common

– Intelligence– Trust– Democracy– Diseases – Disturbance variables

Three Types of SEM

• Classic Econometric• Multiple equations

ζΓxΒyy

ηy

One indicator per latent variable No measurement error

ξx

Classic Econometric

11413432421414 xyyyy

Citationsy3

Quality ratingy4

Publicationsy2

Size of dept.y1

Privatex1

3

2

1

4

β43

β42

β41

β32

β31

γ31

γ41

γ11

11111 xy

Classic Econometric

associations1980

associations1990

democracy1982

trust1980

democracy1991

trust1990

industrialization1980

Noncoreposition

Ethnichomogeneity

2

3

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

Recursive

y2x1

32

y3

y3

y2

x3

2

3

x1

y1

x2

1

Nonrecursivex2 y1

x1

1

y22

y3

y2

x3

2

3

x1 y1

x2

1

Confirmatory Factor Analysis• Latent variables• Measurement error• No causal relationship between latent

variables

δξΛx x x = vector of observed indicatorsΛx = matrix of factor loadingsξ = vector of latent variablesδ = vector of measurement errors

Trust in Individuals

people arehelpful

(x1)


(x2)

people areFair (x3)

1

ξ1

δ1 δ2 δ3

11111 x

21212 x

313 x

Confirmatory Factor Analysis

δξΛx x

λ11 λ21

11

)(00

)(0

)(

3

2

1

VAR

VAR

VAR

General Model

• Includes latent variable model– Relationship between the latent variables

• And measurement model– Relationship between latent variables and

observed variables

General Model

• Latent Variable Model

ζΓξΒηη

η = vector of latent endogenous variablesξ = vector of latent exogenous variablesζ = vector of disturbancesΒ = coefficient matrix for η on η effectsΓ =coefficient matrix for ξ on η effects

General Model

• Measurement Model

εηΛy y δξΛx x

x = indicators of ξΛx = factor loadings of ξ on xy = indicators of ηΛy = factor loadings of η on yδ = measurement error for xε = measurement error for y

General SEM

ε1

y1

ε2

y2

ε3

y3

ε4

y4

ε5

y5

ε6

y6

ε7

y7

ε8

y8

δ1

x1

δ2

x2

δ3

x3

η1

ξ1

η2

ζ1 ζ2β21

γ21

γ11

λ1 λ2 λ3

λ4 λ5 λ6 λ7 λ8 λ9 λ10 λ11

ξ1= industrializationη1 = democracy time 1η2 = democracy time 2x1-x3 = indus. indicator, e.g., energyy1-y4 = democ. indicators time 1y5-y8 = democ. indicators time 2

Six Steps to Modeling

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

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?

Identification

• Are there unique values for parameters?

• Property of model, not data

• 10 = x + y

x = y

2, 8 -1, 11 4, 6

Identification

• Underidentified• Just identified• Overidentified

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

Identification

• Identified? Yes, by 3-indicator rule.

Trust in Individuals

people arehelpful

(x1)


(x2)

people areFair (x3)

1

ξ1

δ1 δ2 δ3

λ11 λ21

Model Fit

• Component Fit– Use Substantive Experience

• Are signs correct?• Any nonsensical results?• R2s for individual equations• Negative error variances?• Standard errors seem reasonable?

Model Fit

• How well does our model fit the data?

• The Test Statistic (Χ2)– T=(N-1)F– df=½(p+q)(p+q+1) - # of parameters

• p = number of y’s• q = number of x’s

– Σ=Σ(θ)– Statistical power

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

mb

mb

dfT

TTIFI

b

mb

T

TTNFI

m

mm

dfN

dfTRMSEA

)1(

Model Fit

Propsenity towardUnconventional

Participation

occupybuilding

d5

1

unofficialstrikes

d4

1

boycotts

d3

1

demon-strations

d2

1

petitions

d1

1

1

Χ2 = 223, df=5, p=.000 IFI = .87 RMSEA = .25

N=801

Respecification

• Theory!– Dimensionality?– Correct pattern of loadings?– Correlated errors of measurement?– Other paths?

• Modification Indexes• Residuals: )θΣ(S ˆ

Propsenity towardUnconventional

Participation

occupybuilding

d5

1

unofficialstrikes

d4

1

boycotts

d3

1

demon-strations

d2

1

petitions

d1

1

1

Respecification

Χ2 = 3.8, df=2, p=.15 IFI = 1.0 RMSEA = .03

N=801

Useful References

• Book from which this talk is drawn: Bollen, Kenneth A. 1989. Structural Equations with Latent Variables. New York: Wiley.

• Ed Rigdon’s website: www.gsu.edu/~mkteer/

• Archives of SEMNET listserv: bama.ua.edu/archives/semnet.html

http://www.gsu.edu/~mkteer/

Structural Equation Modeling: An Overview

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