E1

Structural equation modeling

Rex B Kline Concordia University

Montréal

ISTQL Set E SR models

E2

CFA vs. SR

o Factors:

CFA: Exogenous only

SR: Exogenous + endogenous

E3

CFA vs. SR

o Factors & indicators:

CFA: L → M only

SR: L → M or M → L

E4

Fully latent SR

A B C

1

X1

EX1

1

X2 Y1

EY1

1

Y2

EY2

1

Y3

EY3

1

Y4

EY4

1

DC

1

DB

1 1 1

EX2

1

E5

Partially latent SR (endogenous)

1

A

1

C

1

X1

EX1

1

X2

EX2

Y1

1

Y3

EY3

1

Y4

EY4

DY1

1 1

DC

E6

Partially latent SR (exogenous)

X1 B

1

DB

1

Y2

EY2

1

Y1

EY1

1

C

1

DC

1

Y4

EY4

1

Y3

EY3

1

E7

SR models

o Two parts:

1. Measurement

2. Structural

E8

SR models

o Two steps:

1. Identification

2. Analysis

E9

SR identification

o Two-step rule (sufficient):

1. Measurement as CFA

2. Structural as PA

E10

(c) Structural Model

1 DC

1 DB

A B C

(a) Original SR

Model

1

A

1 1

1

X1

EX1

1

X2

EX2

1

Y1

EY1

1

Y2

EY2

1

Y3

EY3

1

Y4

EY4

1 DC

1 DB

C B

(b) Respecified as a CFA Model

1 1 1

1

X1

EX1

1

X2

EX2

1

Y2

EY2

1

Y3

EY3

1

Y4

EY4

C B A

1

Y1

EY1

E11

SR analysis

o One-step analysis:

A B C

1

X1

EX1 1

X2

1

Y1

EY1 1

Y2

EY2

1

Y3

EY3 1

Y4

EY4

1 DC

1 DB

1 1 1

EX2

E12

SR analysis

o Two-step analysis (fully latent):

1. CFA model

2. SR models

E13

(a) Original SR Model

1

A

1 1

1

X1

EX1

1

X2

EX2

1

Y1

EY1

1

Y2

EY2

1

Y3

EY3

1

Y4

EY4

1 DC

1 DB

C B

(b) Respecified as a CFA Model

1 1 1

1

X1

EX1

1

X2

EX2

1

Y2

EY2

1

Y3

EY3

1

Y4

EY4

C B A

1

Y1

EY1

E14

SR analysis

o R2 effect size:

Indicators

Endogenous factors

E15

Single indicators

o Partially latent (1):

1

A

1

X1

EX1 1

X2

EX2

Y1

DY1 1

1

C

1

Y3

EY3 1

Y4

EY4

1 D C

E16

Single indicators

o Partially latent (2):

1

C

1

Y3

EY3 1

Y4

EY4

D C

1

1

B

1

Y1

EY1 1

Y2

EY2

D B

1

X1

E17

Single indicators

o Requires:

1. Proportion error variance: (1 – rXX) s2

2. Fixed parameters

E18

1

A B

1

C

1

X1

EX1

1

X2

EX2

1

Y1

EY1

1

Y3

EY3

1

Y4

EY4

1

1

DB

1

D C

1

2.30

Ys

E19

1

2.20

Xs

1

B

1

Y1

EY1

1

Y2

EY2

1

DB

1

C

1

Y4

EY4

1

DC

1

Y3

EY3

1

A

1

X1

EX1

E20

X1

X2

Y

1

DY

E21

E2 1

X2

B

1

(1 − 22

r ) 2

2s

E1 1

X1

A

1

(1 − 11r ) 2

1s

EY 1

Y

C

1

(1 − YYr ) 2

Ys

DY 1

E22

Single indicators

o Hayduk, L. A. & Littvay, L. (2012). Should

researchers use single indicators, best indicators, or multiple indicators in structural equation models? BMC

Medical Research Methodology, 12(159). Retrieved from http://www.biomedcentral.com/ 1471-2288/12/159

E23

Acculturation

EGS

1

General Status

1

Acculturation

Scale

EAS

1

Percent Life U.S.

EPL

1

1

Job

EJo

1

Interpersonal

EInt

1

Stress

DSt

1

Depression Scale

DDS

1

SES

1

Education

EEd

1

Income

EInc

1

E24

2 DS.30 s

Acculturation

EGS

1

General Status

1

Acculturation

Scale

EAS

1

Percent Life U.S.

EPL

1

1

Job

EJo

1

Interpersonal

EInt

1

Stress

DSt

1

SES

1

Education

EEd

1

Income

EInc

1

Depression Scale

EDS

1

DDe

1

Depression

1

E25

Exogenous

Direct effects on endogenous Variances Covariances Total

Acc → GS Acc → %Li Acc, SES Acc SES 20

SES → Inc Str → Job E terms (7) GS %Li

Acc → Str Str → Dep D terms (2)

SES → Dep

v = 8; 8(9)/2 = 39

dfM = 39 – 20 = 19

E26

LISREL

title: shen and takeuchi (2001)

error term for depression scale

observed variables

acculscl genstat perlife educ income interper job depscale

latent variables: Accultur Ses Stress Depressi

correlation matrix

1.00

.44 1.00

.69 .54 1.00

.37 .08 .24 1.00

.23 .05 .26 .29 1.00

.12 .08 .08 .08 -.03 1.00

.09 .06 .04 .01 -.02 .38 1.00

.03 .02 -.02 -.07 -.11 .37 .46 1.00

E27

standard deviations

3.119 3.279 2.408 3.270 3.440 2.961 3.604 3.194

sample size is 983

relationships

acculscl = 1*Accultur

genstat perlife = Accultur

educ = 1*Ses

income = Ses

interper = 1*Stress

job = Stress

depscale = 1*Depressi

! depscale as single indicator

Stress = Accultur

Depressi = Ses Stress

E28

set error variance of depscale to 3.06

! fixes the error variance of the single indicator

! rxx = .70, proportion of error variance = .30

! sample variance is 10.200; .30 * 10.200 = 3.06

let the errors of genstat and perlife correlate

path diagram

LISREL output: ND = 3 SC RS

end of program

E29

Reflective vs. formative

E30

Reflective (L→M)

o Contexts:

1. All CFA models

2. Measurement theory

E31

Reflective (L→M)

o Assumes:

1. Interchangeable Ms

2. High, positive rij

3. Unidimensional Ls

E32

Reflective?

o Example:

Income SES

Occupation

Education

Residence

E33

Formative (M→L)

o Assumes:

1. M → L

2. L is a composite

3. L is heterogeneous

E34

Formative (M→L)

o Assumes:

4. Any pattern of rij

5. Ms not interchangeable

E35

(a) L → M block

1

V3

1 E3

V2

1 E2

V1

1 E1

F1

(b) M → L block

V1

1

V2 V3

D

1 F1

E36

Formative (M→L)

o Models with cause indicators:

1. Whole model is SR

2. Identification challenge

3. PLS path modeling

E37

Formative (M→L)

o Identification:

Emit ≥ 2 directs effects

Downstream factors

E38

Acculturation

EGS

1

General Status

1

Acculturation

Scale

EAS

1

Percent Life U.S.

EPL

1

1

Job

EJo

1

Interpersonal

EInt

1

Stress

DSt

1

SES

1

Education

EEd

1

Income

EInc

1

Depression Scale

EDS

1

DDe

1

Depression

1

E39

Formative (M→L)

o Bollen, K. A., & Bauldry, S. (2011). Three Cs in

measurement models: Causal indicators, composite Indicators, and covariates. Psychological Methods, 16, 265–284.

o Diamantopoulos, A. (Ed.). (2008). Formative indicators [Special issue]. Journal of Business Research, 61(12).

o Grace, J. B., & Bollen, K. A. (2008). Representing general theoretical concepts in structural equation models: The role of composite variables. Environmental and

Ecological Statistics, 15, 191–213.

E40

PLS path modeling

o MR and PCA

o Prediction

o Composites (not L)

E41

PLS path modeling

o How to combine variables

o No measurement hypotheses

o No identification issues

E42

PLS path modeling

o Henseler, J., & Wang, H. (2010) (Eds.)

Handbook of partial least

squares: Concepts, methods and

applications. Berlin: Springer-Verlag.

E43

SmartPLS

E44

Other horizons

E45

Variations

o Multiple-samples analysis

o Measurement invariance

E46

Variations

o Millsap, R. E., & Olivera-Aguilar, M. (2012).

Investigating measurement invariance using confirmatory factor analysis. In R. H. Hoyle (Ed.), Handbook of structural equation modeling (pp. 380–392). New York: Guilford Press.

o Nimon, K., & Reio, T., Jr. (2011). Measurement invariance: A foundational principle for quantitative theory building. Human Resource

Development Review, 10, 198–214.

E47

Variations

o Analysis of means

o Latent growth models

E48

Intercept Slope

1

Trial 1

1 E1

Trial 2

1 E2

Trial 4

1 E4

Trial 3

1 E3

Trial 6

1 E6

Trial 5

1 E5

1 1 1

1

1

1 0

1

E49

Variations

o Bollen, K. A., & Curran, P. J. (2006). Latent

curve models: A structural equation

perspective. Hoboken, NJ: Wiley.

o Preacher, K. J., Wichman, A. L., MacCallum, R. C., & Briggs, N. E. (2008). Latent growth curve modeling. Thousand Oaks, CA: Sage.

E50

Variations

o Interactive effects:

Observed variables

Latent variables

E51

Y

DY 1

W

X

XW

M

DM 1

E52

Variations

o Aguinis, H., & Gottfredson, R. K. (2010). Best-practice

recommendations for estimating interaction effects using moderated multiple regression. Journal of Organizational Behavior, 31, 776–786.

doi: 10.1002/job.686. o Klein, A. G., & Muthén, B. O. (2007). Quasi-maximum

likelihood estimation of structural equation models with multiple interaction and quadratic effects. Multivariate Behavioral Research, 42, 647–673.