Structural Equation Modeling

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Structural Equation Modeling

• Kamel Jedidi
• Columbia University

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Outline

• Introduction
• Confirmatory Factor Analysis
• Model Identification
• Model Estimation and Evaluation
• The Full Structural Equation Model
• Summary

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IntroductionWhat are SEM?

• General linear models for describing the
  covariation between observed variables
• Generic mathematical form ??(?) where ? is a
  vector containing all model parameters.
• Regression, Factor Analysis, Simultaneous
  Equation models are special cases

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IntroductionWhat are SEM? Example
Simple regression y ?x ?
Implied Covariance Matrix
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Introduction Why SEM?

• Parsimonious, flexible modeling
• Explicit modeling of latent factors (constructs)
  and measurement error
• Example x1 ?11? ?1
• x2 ?21? ?2
• Resolution of multicollinearity

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IntroductionThe univariate consequences of
measurement error

• x True Score Error ? ?
• ? Var(x) Var(?) Var(?) ? ?
• Thus, Var(x) overestimates the variance of the
  true score

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IntroductionThe bivariate consequences of
measurement error

• A simple regression model with measurement error
• y ?x ? ?

where ?xx is the measurement reliability of x.
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IntroductionThe bivariate consequences of
measurement error

• Impact on goodness-of-fit
• Whats the impact on sample inference?
• Generally, the distortions are not as systematic
  for multiple regression and simultaneous equation
  models

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Confirmatory Factor AnalysisModel
Where x (q ? 1) vector of
indicator/manifest variables ? (n ? 1) vector
of latent constructs (factors) ? (q ? 1) vector
of errors of measurement ? (q ? n) matrix of
factor loadings
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Confirmatory Factor AnalysisExample

• Measures for positive emotions ?1
• x1 Happiness, x2Pride
• Measures for negative emotions ?2
• x3 Sadness, x4Fear
• Model

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Confirmatory Factor AnalysisExample
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Confirmatory Factor AnalysisGraphical
Representation
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Confirmatory Factor AnalysisModel Assumptions
E(?) 0 E(?) 0 Var(?) ? Var(?)
? Cov(?, ?) 0
Implied Mean Vector
Implied Covariance Matrix
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Confirmatory Factor AnalysisExample

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Confirmatory Factor AnalysisModel Identification

• Definition
• The set of parameters ??,?,? is not
  identified if there exists ?1??2 such that ?(?1)
  ?(?2).

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Confirmatory Factor AnalysisIs the one-factor,
two-indicator model identified?

• Example Measures for temperature ? x1
  Celsius, x2Fahrenheit
• Measurement Model
• where ?1 and ?2 are measurement intercepts.

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Confirmatory Factor AnalysisScale indeterminacy

• Recall measurement model
• Origin indeterminacy ? E(?) 0
• Scale (unit) indeterminacy
• How should single-indicator factors be handled?

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Confirmatory Factor AnalysisThe one-factor,
two-indicator model is under identified

• Population covariance matrix
• Implied covariance matrix
• Solution 1 Solution 2

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Confirmatory Factor AnalysisIs the one-factor
three-indicator model identified?
?21
?31
1
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Confirmatory Factor AnalysisThe one-factor
three-indicator model is exactly identified
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• Confirmatory Factor AnalysisIdentification Rules
• - Number of free parameters ? ½ q (q1)
• - Three-Indicator Rule
• n?1
• One non zero element per row of ?
• Three or more indicators per factor
• ? Diagonal

• Two-Indicator Rule
• n gt 1
• ?ij ? 0 for at least one pair i, j, i ? j
• one non-zero element per row of ?
• Two or more indicators per factor
• ? Diagonal

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Confirmatory Factor AnalysisMaximum Likelihood
Estimation
xi i.i.d MVNq(0, ?(?)) i1, , N
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Confirmatory Factor AnalysisOther Estimation
Methods

• Unweighted Least Squares
• Generalized L.S.

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Confirmatory Factor AnalysisThe Asymptotic
Covariance Matrix
Information Matrix
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Confirmatory Factor AnalysisGoodness-of-fit
measures
Root Mean-Square Residual
Correlation Residuals
Goodness-of-Fit Index
Communalities/Reliabilities
Coefficient of Determination

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Confirmatory Factor AnalysisGoodness-of-fit
measures
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Confirmatory Factor AnalysisOther
Goodness-of-fit indices

• Root Mean Square Error of Approximation
• where df (q(q1)/2) t (degrees of
  freedom).
• RMSEA ? 0.05 ? Close fit
• 0.05 lt RMSEA ? 0.08 ? Reasonable fit
• RMSEA gt 0.1 ? Poor fit

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Confirmatory Factor Analysis Multitrait-Multime
thod Example
?x1x2
?x3x1
?x4x3
?x4x2
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Confirmatory Factor Analysis Multitrait-Multime
thod Example
?1
?2
?4
x2
x1
x3
x4
?3
?2
?1
?3
?4
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Brand Halos and Brand Evaluations Lynd Bacon
(1999)
Performance
Quality
Pt2
Pt1
Qd1
Qt1
Pd1
Qt2
Pd2
Qd2
DirtyScooter
TrailBomber
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Brand Halos and Brand EvaluationsSources of
Variance

• Brand Attribute
• DirtyScooter
• Pd1 0.71 0.04
• Pd2 0.74 0.02
• TrailBomber
• Pt1 0.40 0.39
• Pt2 0.41 0.30

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Convergent and Discriminant ValidityBagozzi and
Yi (1993)

• Attitude towards coupons (?1) with three semantic
  differential measures x1pleasant/unpleasant
• x2good/bad
• x3favorable/unfavorable
• Subjective norms (?2) with two measures
• x4 Most people who are important to me think I
  definitely should use coupons for
  shopping in the supermarket
• x5 Most people who are important to me
  probably consider my use of coupons to be
  wise.

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Convergent and Discriminant Validity Bagozzi and
Yi (1993)
.82
x2
?2 .33
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Convergent and Discriminant ValidityBagozzi and
Yi (1993)

• Convergent validity
• - Goodness-of-fit
• - All loadings are high and significant
• Discriminant validity H0 ?1 is rejected
• Measurement reliability (x1.56, x2.67, x3.53,
  x4.48, x5.81)

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The Full Structural Equation Model Measurement
Model

• Where
• x (q ? 1) vector of exogenous
  indicator/manifest variables
• y (p ? 1) vector of endogenous
  indicator/manifest variables
• ? (n ? 1) vector of exogenous latent constructs
  with mean 0 and variance ??
• ? (m ? 1) vector of endogenous latent
  constructs
• ? (q ? 1) vector of errors of measurement with
  mean 0 and variance ??
• ? (p ? 1) vector of errors of measurement with
  mean 0 and variance ??
• ?x (q ? n) matrix of factor loadings
• ?y (p ? m) matrix of factor loadings

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The Full Structural Equation Model Structural
Model

• where
• B (m x m) Coefficient Matrix for the effect of
  ? on ?
• ? (m x n) Coefficient Matrix for the effect ?
  on ?
• ? (m x 1) Vector of errors, E(?) 0 , COV(?,
  ?) ? , COV(?, ?) 0

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The Full Structural Equation Model The Implied
Covariance Matrix

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The Full Structural Equation Model Identification

• Number of parameters lt(pq)(pq1)/2
• Two-Step Rule
• - Measurement Model Identification
• - Structural Model Identification

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The Full Structural Equation Model Structural
Model Identification

• Null B Rule (B0)
• Recursive Rule
• - B Triangular
• - ? Diagonal
• Order Condition
• ith equation is identified if of variables
  excluded from ith equation is ? m-1
• Rank Condition
• - Form
• - ith equation is identified if rank of Ci m
  1 where Ci formed from those columns of C
  that have 0 in the ith row.

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The Full Structural Equation Model Structural
Model Identification Example
?1
?1
?1
?2
?2
?2
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The Full Structural Equation ModelStructural
Model Identification Example

• Form
• Rank of is m-12-11
• Rank of is m-12-11
• Both equations are identified

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Construct Validation by Use of Panel Model
Bagozzi and Yi (1993)
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Construct Validation by Use of Panel Model
Bagozzi and Yi (1993)

• ?31 and ?42 capture temporal stability
• ?21 and ?43 reflect discriminant validity
• Convergent validity is assessed by overall model
  fit and by the magnitude and significance of the
  factor loadings
• The covariance between two serially correlated
  errors is a measure of specific variance

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Construct Validation by Use of Panel Model
Bagozzi and Yi (1993)
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Construct Validation by Use of Panel Model
Bagozzi and Yi (1993)

• Convergent validity
• - Goodness-of-fit
• - All loadings are high and significant
• - Factorial invariance holds
• Discriminant validity H0 ?21 1 and ?43 1 are
  rejected
• Temporal stability and

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Structural Equation ModelingExtensions

• Multi-group analysis
• Useful for testing moderating effects
• Finite mixture SEM
• Useful for segmentation and uncovering
  unobserved moderators
• Hierarchical Bayesian SEM
• Useful for capturing observed and unobserved
  heterogeneity

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Summary

• SEM is a powerful tool for modeling latent
  variables and treating measurement error
• Make sure to check for identification prior to
  estimation
• Model EvaluationRMSEA works best
• Available software
• Proc CALIS (SAS)
• AMOS (SPSS)
• LISREL