structural equation models - pennsylvania state university · 2019-02-03 · evaluating structural...

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]


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


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


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]

Model modification and model comparison [see

Appendix E]

□ Measurement model

□ Latent variable model



Local fit measures[see Appendix F on how to obtain robust chi-square

statistics and standard errors; another option is to use

bootstrapping]

Measurement model

□ factor loadings, factor (co)variances, and error

variances

□ Reliability/convergent validity and discriminant validity

[composite reliability can be computed introducing

constrained additional parameters]



Latent variable model

□ structural coefficients and equation disturbances

□ direct, indirect, and total effects [see Appendix G]

□ explained variation in endogenous constructs

Power [see Appendix H]

Model-based residual analysis

Cross-validation

Model equivalence and near equivalence [see

Appendix I]

Latent variable scores [see Appendix J]


focalconstruct

Reflective vs. formative measurement models

focalconstruct

focalconstruct focal

construct


Criteria for distinguishing between

reflective and formative indicator models

Are the indicators manifestations of the underlying

construct (rather than 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.


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


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

normal theory WLS 2

S-B scaled 2

2 corrected for non-

normality

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


known - random

population covariance matrix

0

0

~

best fit of the model to 0

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)

Types of error in covariance structure modeling


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;

nBFt

BFnBFor

tGF

nGFt

GF • type I indices:

• type II indices:)()(

tBFE

nBF

tBF

nBF

or

nGF

tGFE

nGF

tGF


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 2

lowest df

highest 2

highest df


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

dire

ct

indire

ct


Decision

True state of nature

Accept H0

H0 true H0 false

Reject H0

Correct

decision

Correct

decision

Type I

error (a)

Type II

error (b)


test statistic

power

non-

significant

significant

low high


η1

η2

η3

η4

η5

η1

η1

η1

η2

η2

η2

η3

η3

η3

η4

η4

η4

η5

η5

η5

structural equation models - pennsylvania state university · 2019-02-03 · evaluating structural...

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