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.