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