rex b kline - concordia university · cfa respecify o residual patterns: result correlation...
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D1
o
Structural equation modeling
Rex B Kline Concordia University
Montréal
ISTQL Set D CFA models
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D2
Resources o Bollen, K. A., & Hoyle, R. H. (2012). Latent variable
models in structural equation modeling. In R. H. Hoyle (Ed.), Handbook of structural equation
modeling (pp. 56–67). New York: Guilford. o Fabrigar, L. R., & Wegener, D. T. (2012). Exploratory
factor analysis. New York: Oxford University Press. o Kline, R. B. (2013b). Exploratory and confirmatory
factor analysis. In Y. Petscher & C. Schatsschneider (Eds.), Applied quantitative analysis in the social
sciences (pp. 171–207). New York: Routledge.
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D3
EFA
o Phases:
1. Specification
2. Extraction
3. Retention
4. Rotation
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D4
Extraction methods
1. Principle components analysis (PCA)
2. Principle axis factoring (PAF)
3. Alpha factoring
4. ML factoring
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D5
PCA
B A
X4 X5 X6 X3 X2 X1
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D6
PAF
A B
X6
E6
X5
E5
X4
E4
X1
E1
X2
E2
X3
E3
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D7
Indicator variance
1 − rXX
Error
Unique
Common Specific
Systematic
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D8
EFA
o Retention:
No need to specify
But best by theory
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D9
EFA
o Retention:
Parallel analysis
Scree plots
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D10
4
2
0
3
1
1 2 3 4 5 6 7 8
Factor
Eig
en
va
lue
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D11
EFA
o Rotation:
1. Orthogonal
2. Oblique
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D12
EFA
o Orthogonal:
1. Varimax
2. Quartimax
3. Equamax
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D13
EFA
o Oblique:
1. Promax
2. Oblimin
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D14
EFA
o Rotation:
Infinite
Not identified
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D15
a) EFA (unrestricted; rotation)
A B
X6
E6
X5
E5
X4
E4
X1
E1
X2
E2
X3
E3
1
1
X1
E1
1
X2
E2
1
X3
E3
A
1
1
X4
E4
1
X5
E5
1
X6
E6
B
b) CFA (restricted; no rotation)
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D16
CFA after EFA
o Does not “confirm” EFA:
Restricted vs. unrestricted
Items are “noisy”
Follow EFA with EFA
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D17
CFA after EFA o Osborne, J. W., & Fitzpatrick, D. C. (2012). Replication
analysis in exploratory factor analysis: What it is and why it makes your analysis better. Practical
Assessment, Research & Evaluation, 17. Retrieved from http://pareonline.net/pdf/ v17n15.pdf
o van Prooijen, J.-W., & van der Kloot, W. A. (2001). Confirmatory analysis of exploratively obtained factor structures. Educational and Psychological
Measurement, 61, 777–792.
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D18
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D19
Background
1
EE
Ethnic Age
1
EA
Gender
1
EG
1
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D20
A
1
E3
X3 X2
1
E2
X1
1
E1
1 +
−
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D21
CFA specification
o Standard model:
Continuous indicators (X)
A → X ← E
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D22
Reflective measurement
X T E= +
2 2 2
X T Eσ = σ + σ
2
2
σ=
σT
XX
X
r
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D23
Reflective measurement
1−XXr
but rXX estimates a single source
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D24
CFA specification
o Standard model:
Independent E
A B
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D25
CFA specification
o Unidimensional:
Simple indicator (A → X only)
No Ei Ej
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D26
CFA specification
o Unidimensional:
Precise test
Convergent validity
Discriminant validity
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D27
CFA specification
o Multidimensional:
Complex indicator
Ei Ej
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D28
CFA specification
o Ei Ej:
Indicators share something
Repeated measures
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D29
CFA specification
o Multidimensional caution:
Increases complexity
“Cheap” way to improve fit
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D30
CFA specification
o Special variations:
Hierarchical CFA
MTMM models
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D31
g
1
1
X4
E4
1
X5
E5
1
X6
E6
Visual-
Spatial
1
DVS
1
1
X1
E1
1
X2
E2
1
X3
E3
1
DVe
1
1
X7
E7
1
X8
E8
1
X9
E9
Memory
1
DMe
1
Verbal
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D32
X4 X5 X6 X1 X2 X3 X7 X8 X9
1 1 1
Trait 1 Trait 2 Trait 3
1 1 1
Method 1 Method 2 Method 3
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D33
Trait 2 Trait 1 Trait 3
1 1 1
X1
1
E1
X2
1
E2
X3
1
E3
X4
1
E4
X5
1
E5
X6
1
E6
X7
1
E7
X8
1
E8
X9
1
E9
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D34
CFA specification
o Eid, M., Nussbeck, F. W., Geiser, C., Cole,
D. A., Gollwitzer, M., & Lischetzke, T. (2008). Structural equation modeling of multitrait-multimethod data: Different models for different types of methods. Psychological Methods, 13, 230–253.
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D35
CFA identification
o Necessary:
dfM ≥ 0
Scale each latent
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D36
Scale E
ULI constraint:
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D37
Scale factor 1. Reference (marker) variable
ULI = 1, unstandardized
2. Standardize factors UVI = 1
3. Effects coding AVE = 1, all same metric
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D38
1
1
X1
E1
1
X2
E2
1
X3
E3
A
1
1
X4
E4
1
X5
E5
1
X6
E6
B
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D39
1
1
X4
E4
1
X5
E5
1
X6
E6
B 1
1
X1
E1
1
X2
E2
1
X3
E3
A
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D40
1 2 3 13
λ + λ + λ=
λ1
1
X1
E1
1
X2
E2
1
X3
E3
A
λ3 λ2
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D41
1 2 3 13
λ + λ + λ=
1 2 33λ = − λ − λ
2 1 33λ = − λ − λ
3 1 23λ = − λ − λ
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D42
CFA identification
o Counting parameters:
1. Exog: Vars. + Covs.
2. Endog: Direct effects
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D43
CFA identification
o Standard models:
1 factor, ≥ 3 indicators
≥ 2 factors, ≥ 2 indicators
But…
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D44
CFA identification
o Nonstandard models:
No single heuristic
Undecidable
Ambiguous status
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D45
TABLE 6.1. Identification Rule 6.6 for Nonstandard Confirmatory Factor Analysis Models with
Measurement Error Correlations
For a nonstandard CFA model with measurement error correlations (Rule 6.6)
to be identified, all three of the conditions listed next must hold:
For each factor, at least one of the following must hold: (Rule 6.6a)
1. There are at least three indicators whose errors are uncorrelated with each other.
2. There are at least two indicators whose errors are uncorrelated and either
a. the errors of both indicators are not correlated with the error term of a third
indicator for a different factor, or
b. an equality constraint is imposed on the loadings of the two indicators.
For every pair of factors, there are at least two indicators, one from (Rule 6.6b)
each factor, whose error terms are uncorrelated.
For every indicator, there is at least one other indicator (not necessarily (Rule 6.6c)
of the same factor) with which its error term is not correlated.
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D46
For each factor, at least one of the following must hold: (Rule 6.6a)
1. There are at least three indicators whose errors are uncorrelated with each other.
2. There are at least two indicators whose errors are uncorrelated and either
a. the errors of both indicators are not correlated with the error term of a third
indicator for a different factor, or
b. an equality constraint is imposed on the loadings of the two indicators.
(c)
X1
EX1
1
X2
EX2
1
A
1
X3
EX3
1
X4
EX4
1
B
1
X1
EX1
1
X2
EX2
1
A
1
X3
EX3
1
X4
EX4
1
B
1
(d)
� �
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D47
TABLE 6.2. Identification Rule 6.7 for Multiple Loadings of Complex Indicators in Nonstandard
Confirmatory Factor Analysis Models and Rule 6.8 for Error Correlations of Complex Indicators
Factor loadings
For every complex indicator in a nonstandard CFA model: (Rule 6.7)
In order for the multiple factor loadings to be identified, both
of the following must hold:
1. Each factor on which the complex indicator loads must satisfy
Rule 6.6a for a minimum number of indicators.
2. Every pair of those factors must satisfy Rule 6.6b that each
factor has an indicator that does not have an error correlation
with a corresponding indicator on the other factor of that pair.
Error correlations
In order for error correlations that involve complex indicators (Rule 6.8)
to be identified, both of the following must hold:
1. Rule 6.7 is satisfied.
2. For each factor on which a complex indicator loads, there must be
at least one indicator with a single loading that does not have an
error correlation with the complex indicator.
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D48
CFA estimates
o Unstandardized:
1. Indicators loadings (B)
2. Factor, error variances
3. Factor, error covariances
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D49
CFA estimates
o Standardized:
1. Indicators loadings (r, b)
2. Proportion unexplained
3. Factor, error correlations
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D50
CFA estimates
o Failure to converge:
1. Data matrix (NPD)
2. Poor start values
3. Small N, 2 ind./factor
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D51
CFA estimates
o Heywood cases (inadmissible):
1. Error variance < 0
2. | r or R2 | > 1.0
3. NPD parameter matrix
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D52
1 1
Family of Origin
Marital
Adjustment
Father
1
EFa
Mother
1
EMo
Father- Mother
1
EFM
1
EIn
Intimacy
1
EPr
Problems
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D53
Group 2: Wives
THETA-DELTA
problems intimacy father mother fa_mo
-------- -------- -------- -------- --------
problems 520.305
(130.844)
3.977
intimacy - - -27.093
(104.927)
-0.258
father - - - - 32.147
(29.214)
1.100
mother - - - - 9.967 63.416
(26.870) (28.138)
0.371 2.254
fa_mo - - - - - - - - 97.049
(25.232)
3.846
Squared Multiple Correlations for X - Variables
problems intimacy father mother fa_mo
-------- -------- -------- -------- --------
0.520 1.052 0.821 0.661 0.531
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D54
CFA estimates
o Heywood causes:
Identification
Poor start values
Small N, 2 inds./factor
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D55
CFA analysis
o Testing strategy:
1. Fit 1-factor model
2. Nested under higher-order
3. Compare with 2
Dχ
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D56
1
ETr
Triangles
1
Spatial Memory
ESM
1
Matrix Analogies
EMA
1
Gestalt Closure
EGC
1
Photo Series
EPS
1
EHM
Hand Movements
1
Number Recall
ENR
1
Word Order
EWO
1
General
1 1 1 1 1 1 1 1
1 1
Sequential Processing
EHM
Hand Movements
Number Recall
ENR
Word Order
EWO
Simultaneous Processing
ETr
Triangles Spatial
Memory
ESM
Matrix Analogies
EMA
Gestalt Closure
EGC
Photo Series
EPS
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D57
CFA analysis
o Example: 4-factor model:
4 vs. 3
4 vs. 2
4 vs. 1
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D58
CFA respecify
o Options:
1. Number of factors
2. Indicator-factor match
3. Error correlations
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D59
CFA respecify
o Residual patterns:
Result Correlation residuals Respecification
Indicator has low standardized loading on original factor
High correlation residuals with indicators of another factor
Switch loading of indicator to other factor
Indicator has reasonably high standardized loading on original factor
High correlation residuals with indicators of another factor
Allow indicator to also load on the other factor Allow measurement errors to covary
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D60
CFA respecify
o Wrong number of factors:
Discriminant validity
Convergent validity
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D61
CFA respecify
o MIs in latent variable models:
Approach with caution
Nonsensical respecification
May not be identified
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D62
Observations = v (v + 1)/2 = 36
Parameters = 17
dfM = 19
1 1 1 1 1 1 1 1
1 1
Sequential Processing
EHM
Hand Movements
Number Recall
ENR
Word Order
EWO
Simultaneous Processing
ETr
Triangles Spatial
Memory
ESM
Matrix Analogies
EMA
Gestalt Closure
EGC
Photo Series
EPS
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D63
Exogenous variables
Direct effects on endogenous variables Variances Covariances Total
Sequential → NR Sequential → WO Seq, Sim Seq Sim 17
Simultaneous → Tr Simultaneous → SM E terms (8)
Simultaneous → MA Simultaneous → PS
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D64
Example
o Amos
o EQS o lavaan
o LISREL
o Mplus
o Stata
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D65
title: principles and practice of sem (4th ed.), rex kline
two-factor model of the kabc-i, figure 9.7, table 13.1
data: file is "kabc-mplus.dat";
type is stdeviations correlation;
nobservations = 200;
variable: names are handmov numbrec wordord gesclos triangle spatmem
matanalg photser;
analysis: type is general;
model:
Sequent by handmov numbrec wordord;
Simul by gesclos triangle spatmem matanalg photser
! first indicator in each list is automatically
! specified as the reference variable
output: sampstat modindices(all, 0) residual standardized tech4;
! requests sample data matrix, residual diagnostics,
! modification indexes > 0, all standardized
! solutions (STDYX is reported), and estimated
! correlation matrix for all variables
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D66
3.40 2.40 2.90 2.70 2.70 4.20 2.80 3.00
1.00
.39 1.00
.35 .67 1.00
.21 .11 .16 1.00
.32 .27 .29 .38 1.00
.40 .29 .28 .30 .47 1.00
.39 .32 .30 .31 .42 .41 1.00
.39 .29 .37 .42 .58 .51 .42 1.00
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D67
CFA indicators
o Indicators:
Scale: Default ML
Likert: Other method
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D68
CFA indicators
o Item distributions:
1. Binary (e.g., T / F)
2. Likert (3-6)
3. Likert (≥ 7)
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D69
CFA indicators
o Estimation options:
1. Corrected ML:
a. Robust SEs
b. Santorra-Bentler
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D70
CFA indicators
o Estimation options:
2. Robust WLS:
a. Item thresholds
b. Latent response variable
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D71
CFA indicators
o Threshold:
Location on latent dimension
Differentiates categories
Estimated as z
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D72
Example: 1 = disagree
2 = not sure
3 = agree
−1.62 1.15
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D73
A
1X*
1 1X
E *
2X*
1 2X
E *
3X*
1 3X
E *
X1 X2 X3
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D74
CFA indicators
o Latent response variables:
Sample polychoric
Predicted polychoric
Correlation residuals
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D75
CFA indicators
o Estimation options:
3. ML + numerical integration
a. ↑ computation b. Markov chain Monte
Carlo
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D76
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D77
CFA indicators
o Estimation options:
4. IRT, ICC
a. Difficulty, discrimination
b. Logit, probit link
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D78
3.0 2.0 1.0 0 −1.0 −2.0 −3.0
Latent Ability (θ)
.9
.8
.7
.6
.5
.4
.3
.2
.1
0
1.0
Pro
ba
bili
ty o
f C
orr
ec
t
Re
spo
nse
tangent line
difficulty
ICC
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D79
CFA indicators
o Estimation options:
5. Bootstrapping:
a. Very biased small N
b. Not as developed
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D80
CFA indicators
o Estimation options:
6. Create parcels:
a. Homogenous item set
b. Total score
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D81
● ● ●
1
A
It 1 It 2 It 33 ● ● ●
1
B
It 66 It 34 It 35 ● ● ●
1
It 99 It 67 It 68
C
A
1
Pr 1 (It 1–It 11)
Pr 2 (It 12–It 22)
Pr 3 (It 23–It 33)
B
1
Pr 4 (It 34–It 44)
Pr 5 (It 45–It 55)
Pr 6 (It 26–It 66)
C
1
Pr 8 (It 78–It 88)
Pr 9 (It 89–It 99)
Pr 7 (It 67–It 77)
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D82
Cautions about parcels
1. Assumes unidimensional
2. Ways to parcel
3. Mask multidimensionality
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D83
CFA indicators
o Edwards, M. C., Wirth, R. J., Houts, C. R., & Xi, N.
(2012). Categorical data in the structural equation modeling framework. In R. Hoyle (Ed.), Handbook of structural equation
modeling (pp. 195–208). New York: Guilford Press.
o Wirth, R. J., & Edwards, M. C. (2007). Item factor analysis: Current approaches and future directions. Psychological Methods, 12, 58–79.
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D84
CFA indicators
o Bernstein, I. H., & Teng, G. (1989). Factoring items and
factoring scales are different: Spurious evidence for multidimensionality due to item categorization. Psychological Bulletin, 105, 467–477.
o Bandalos, D. L., & Finney, S. J. (2001). Item parceling issues in structural equation modeling. In G. A. Marcoulides and R. E. Schumaker (Eds.), New
developments and techniques in structural
equation modeling (pp. 269–296). Mahwah, NJ: Erlbaum.
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D85
Exploratory SEM
o CFA-EFA-SR hybrid
o Restricted + unrestricted
o EFA part is rotated
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D86
Y6 Y5 Y4 Y3 Y1 Y2
1 1
DF 1
DC
EY1
1 EY2
1 EY3
1 EY4
1 EY5
1 EY6
1
C F
X1 EX1 1
EX2 1
X2
X3 EX3 1
X4 EX4 1
X5 EX5 1
X6 EX6 1
1
A
1
B
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D87
Exploratory SEM
o Marsh, H. W., Morin, A. J. S., Parker, P. D., &
Kaur, G. (2014). Exploratory structural equation modeling: Integration of the best features of exploratory and confirmatory factor analysis. Annual
Review of Clinical Psychology, 10, 85–110.