Evaluation of
structural equation models
Hans Baumgartner
Penn State University
Evaluating structural equation models
Issues related to the initial specification
of theoretical models of interest
 Model specification:
□
Measurement model:



EFA vs. CFA
reflective vs. formative indicators [see Appendix A]
number of indicators per construct [see Appendix B]
total aggregation model
 partial aggregation model
 total disaggregation model

□
Latent variable model:


recursive vs. nonrecursive models
alternatives to the target model [see Appendix C for an
example]
Evaluating structural equation models
x1
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x7
x8
d1
d2
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d4
d5
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d7
d8
Evaluating structural equation models
z1
z2
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Evaluating structural equation models
Criteria for distinguishing between
reflective and formative indicator models
 Are the indicators manifestations of the underlying
construct or defining characteristics of it?
 Are the indicators conceptually interchangeable?
 Are the indicators expected to covary?
 Are all of the indicators expected to have the same
antecedents and/or consequences?
Based on MacKenzie, Podsakoff and Jarvis,
JAP 2005, pp. 710-730.
Evaluating structural equation models
Consumer Behavior
Attitudes
Aad as a mediator of advertising effectiveness:
Four structural specifications (MacKenzie et al. 1986)
Affect transfer hypothesis
Cad
Aad
Cb
Ab
BI
Reciprocal mediation hypothesis
Cad
Aad
Cb
Ab
BI
Dual mediation hypothesis
Cad
Aad
Cb
Ab
BI
Independent influences hypothesis
Cad
Aad
Cb
Ab
BI
Evaluating structural equation models
Issues related to the initial specification
of theoretical models of interest
 Model misspecification
□ omission/inclusion of (ir)relevant variables
□ omission/inclusion of (ir)relevant relationships
□ misspecification of the functional form of
relationships
 Model identification
 Sample size
 Statistical assumptions
Evaluating structural equation models
Data screening
 Inspection of the raw data
□
□
□
detection of coding errors
recoding of variables
treatment of missing values
 Outlier detection
 Assessment of normality
 Measures of association
□
□
□
regular vs. specialized measures
covariances vs. correlations
non-positive definite input matrices
Evaluating structural equation models
Model estimation and testing
 Model estimation
 Estimation problems
□
□
□
□
nonconvergence or convergence to a local optimum
improper solutions
problems with standard errors
empirical underidentification
 Overall fit assessment [see Appendix D]
 Local fit measures
[see Appendix E on how to obtain robust standard
errors]
Evaluating structural equation models
Overall fit indices
Stand-alone fit indices
Incremental fit indices
Information
2 test and Noncentralitybased
theory-based
Others
variations
minimum fit
function 2
(C1)
NCP
AIC
(S)RMR
NFI
[2 or f]
IFI
Rescaled
NCP (t)
SBC
GFI
RFI
[2/df]
TLI
CIC
PGFI
ECVI
AGFI
CFI
[2-df]
Gamma
hat
TLI
[(2-df)/df]
RMSEA
S-B scaled 2
(C3)
MC
2/df
minimum fit
function f
Scaled LR
Type II indices
measures
normal theory
WLS 2 (C2)
2 corrected
for nonnormality
(C4)
Type I indices
measures
CN
Evaluating structural equation models
Types of error in covariance structure modeling
best fit of the model to S
for a given discrepancy function
̂
known - random
~

0
error of approximation
(an unknown constant)

0
unknown - fixed
unknown - fixed
best fit of the model to 0
for a given discrepancy function
population covariance matrix
Evaluating structural equation models
Incremental fit indices
• type I indices:
• type II
GFt GFn
or
GFt
GF  GFn
t
indices:
E (GF )  GFn
t
GFt, BFt
=
GFn, BFn
=
E(GFt), E(BFt) =
BFn  BFt
BFn
or
BFn  BF
t
BFn  E ( BF )
t
value of some stand-alone goodness- or
badness-of-fit index for the target model;
value of the stand-alone index for the
null model;
expected value of GFt or BFt assuming that
the target model is true;
Evaluating structural equation models
Model estimation and testing
 Measurement model
□ factor loadings, factor (co)variances, and error
variances
□ reliabilities and discriminant validity
 Latent variable model
□ structural coefficients and equation disturbances
□ direct, indirect, and total effects [see Appendix F]
□ explained variation in endogenous constructs
Evaluating structural equation models
Direct, indirect, and total effects
-.28
inconveniences
direct
rewards
.44
encumbrances
Aact
indirect
-.31
rewards
.49
B
-.15
.48
BI
-.05
.24
B
-.03
inconveniences
-.15
-.28
rewards
encumbrances
BI
-.05
inconveniences
encumbrances
1.10
-.05
.44
-.31
Aact
.48
-.05
-.03
BI .24
B
Evaluating structural equation models
Model estimation and testing
 Power [see Appendix G]
 Model modification and model comparison [see
Appendix H]
□ Measurement model
□ Latent variable model
 Model-based residual analysis
 Cross-validation
 Model equivalence and near equivalence [see
Appendix I]
 Latent variable scores [see Appendix J]
Evaluating structural equation models
True state of nature
Accept H0
H0 true
H0 false
Correct
decision
Type II
error (b)
Decision
Reject H0
Type I
error (a)
Correct
decision
Evaluating structural equation models
power
low
nonsignificant
test statistic
significant
high
Evaluating structural equation models
Model comparisons
saturated structural model (Ms)
lowest 2
lowest df
next most likely unconstrained model (Mu)
target model (Mt)
next most likely constrained model (Mc)
null structural model (Mn)
highest 2
highest df
Evaluating structural equation models
η3
η1
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