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Evaluation of structural equation models. Hans Baumgartner Penn State University. 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] - PowerPoint PPT Presentation
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Evaluation of structural equation models
Hans BaumgartnerPenn 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
d1 d2 d3 d4 d5 d6 d7 d8
x1 x2 x3 x4 x5 x6 x7 x8
x1 x2
Evaluating structural equation models
x1 x2 x3 x4 x5 x6 x7 x8
x1 x2
z1 z2
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 BehaviorConsumer BehaviorAttitudes
Aad as a mediator of advertising effectiveness:Four structural specifications (MacKenzie et al. 1986)
Cb
Cad Aad
Ab BI
Cb
Cad Aad
Ab BI
Cb
Cad Aad
Ab BI
Cb
Cad Aad
Ab BI
Affect transfer hypothesis
Reciprocal mediation hypothesis
Dual mediation hypothesis
Independent influences hypothesis
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
Type I indices Type II indices
NFI
RFI
IFI
TLI
[2 or f]
[2/df]
CFI [2-df]
TLI[(2-df)/df]
2 test andvariations
Noncentrality-based
measures
Information theory-based
measuresOthers
minimum fit function 2
(C1)
normal theory WLS 2 (C2)
S-B scaled 2
(C3)
2 corrected for non-
normality (C4)
2/df
minimum fit function f
Scaled LR
NCP
Rescaled NCP (t)
RMSEA
MC
AIC
SBC
CIC
ECVI
(S)RMR
GFI
PGFI
AGFI
Gamma hat
CN
Evaluating structural equation models
known - random
population covariance matrix
0
0~
best fit of the model to S0
for a given discrepancy function
unknown - fixedunknown - fixed
best fit of the model to S
for a given discrepancy function
error of approximation(an unknown constant)
error of estimation
(an unknown random variable)
over
all e
rror
(an
unkn
own
rand
om v
aria
ble)
Types of error in covariance structure modeling
Evaluating structural equation models
Incremental fit indices
GFt, BFt = value of some stand-alone goodness- or badness-of-fit index for the target model;
GFn, BFn = value of the stand-alone index for the null model;
E(GFt), E(BFt) = expected value of GFt or BFt assuming that the target model is true;
nBFtBFnBF
ortGF
nGFtGF • type I indices:
• type II indices:)()(
tBFEnBF
tBFnBF
ornGF
tGFE
nGFt
GF
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
inconveniences
rewards
encumbrances
Aact BI B
-.28
.44
-.05
1.10 .49
inconveniences
rewards
encumbrances
BI B.24
inconveniences
rewards
encumbrances
Aact BI B-.28
.44
-.05
.48 .24
-.31
-.05
.48
-.15
-.03
-.31
-.05
-.15
-.03
directindirect
total
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
Decision
True state of nature
Accept H0
H0 true H0 false
Reject H0
Correctdecision
Correctdecision
Type I error ( )a
Type II error ( )b
Evaluating structural equation models
test statistic
power
non-significant
significant
low high
Evaluating structural equation models
Model comparisons
saturated structural model (Ms)
null structural model (Mn)
target model (Mt)
next most likely unconstrained model (Mu)
next most likely constrained model (Mc)
lowest c2
lowest df
highest c2
highest df
Evaluating structural equation models
η1
η2
η3
η4
η5
η1
η1
η1
η2
η2
η2
η3
η3
η3
η4
η4
η4
η5
η5
η5
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