Evaluation of structural equation models - PowerPoint Presentation

Slide1

Evaluation of structural equation models

Hans BaumgartnerPenn State University

Issues related to the initial specification of theoretical models of interestModel specification:Measurement model:EFA vs. CFAreflective vs. formative indicators [see Appendix A]number of indicators per construct [see Appendix B]

total aggregation modelpartial aggregation modeltotal disaggregation modelLatent variable model:recursive vs.

nonrecursive

models

alternatives to the target model [see Appendix C for an example]

d

1

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x

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2

Criteria for distinguishing between reflective and formative indicator modelsAre 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.

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 screeningInspection of the raw data detection of coding errorsrecoding of variablestreatment of missing valuesOutlier detection

Assessment of normalityMeasures of associationregular vs. specialized measurescovariances vs. correlationsnon-positive definite input matrices

Model estimation and testingModel estimationEstimation problemsnonconvergence or convergence to a local optimumimproper solutions

problems with standard errorsempirical underidentificationOverall fit assessment [see Appendix D]Local fit measures

[see Appendix E on how to obtain robust standard errors]

known - random

population covariance matrix

best fit of the model to

S

0

for a given discrepancy function

unknown - fixed

unknown - 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)

overall error

(an unknown random variable)

Types of error in covariance structure modeling

Incremental fit indicesGFt, BFt

= value of some stand-alone goodness- or badness-of-fit index for the target model;GFn

, BF

n

= value of the stand-alone index for the null model;

E(GF

t

), E(BF

t

) = expected value of GF

t

or BF

t

assuming that

the target model is true;

type I indices:

type II indices:

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

Direct, indirect, and total effectsinconveniences

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

direct

indirect

total

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]

Decision

True state of nature

Accept H

0

H

0

true

H

0

false

Reject H

0

Correct

decision

Correct

decision

Type I

error (

a)

Type II

error (

b)

test statistic

power

non-

significant

significant

low

high

Model comparisons

saturated structural model (Ms)null structural model (M

n

)

target model (M

t

)

next most likely unconstrained model (M

u

)

next most likely constrained model (M

c

)

lowest

c

2

lowest df

highest

c

2

highest df