Download - Structural Equation Modeling
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Structural Equation Modeling
Mgmt 291Lecture 3 – CFA and Hybrid Models
Oct. 12, 2009
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Measurement is Everything Nothing can be done with wrong or
unreliable measurements.
“Measurement is Everything”.
In research presentation or paper submission, measurement is the part being challenged the most.
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Latent Variables are everywhere in research “The true power of SEM comes from latent
variable modelling “ “Variables in psychology and other social
science are rarely (never?) measured directly” the effects of the variable are measured
Intelligence, self-esteem, depression Reaction time, diagnostic skill Democracy, Socio-Economical Status Legitimacy, Management Skill (soul, angels, … - hypothetical construct)
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Beyond Validity and Reliability:Between concept and indicators
Validity: Measures what it intends to measure.
Reliability: Consistency
Precision
repeatability
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Latent Vars and Observed Indicators
What to be studied is:
L1
X3.1 X4.1 X5.1X2.1X1.1
L2
X3.2 X4.2 X5.2X2.2X1.2
LatentVariable
Indicator3Indicator1
vs.Latent Observed
E 1.1 E 2.1 E 3.1 E 4.1 E 5.1 E 1.2 E 2.2 E 3.2 E 4.2 E 5.2
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Exploratory Factor Analysis
SPSSFor DataReductionFactor analysis
GIGO
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Confirmative Factor Analysis
1) Equations & Diagrams: model representation
2) Identification & Estimation 3) Errors and Evaluation:
assumptions & fit indexes 4) Explanation
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1) Equations & Diagrams: model representation X 1.1 = Ø1 L1 + e 1.1
X 2.1 = Ø2 L1 + e 2.1
X 3.1 = Ø3 L1 + e 3.1
X 4.1 = Ø4 L1 + e 4.1
Loadings - Ø1 … X ~ similar to endogenous variables L ~ similar to exogenous variables
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More on Equations
X = L+ e
Measured
True Score
Error
Relationship between Measured <–> true score Observed <–> latent variable Indicator <–> construct or factor
Uniquefactor
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Diagram representation
L1
X3.1 X4.1 X5.1X2.1X1.1
L2
X3.2 X4.2 X5.2X2.2X1.2
E1.1E4.1 E5.1E3.1E2.1
E1.2 E2.2 E3.2 E4.2 E5.2
Knowing SEM
Research Presentation
Assignment Report
Classroom Participation
e1
e2
e3
Co-vary
X 1.1 ~ X 5.1 load on L 1
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Types of Measurement Models
Uni-dimensional (each indicator loads only on one factor, error terms independent from each other)
Multi-dimensional
Single-factor Multifactors
L1
X3.1 X4.1X2.1X1.1
L2
X3.2 X4.2X2.2
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Non-recursive Type
EducationIncome
Occupation
Socio-economical Status
?
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2) Identification and Estimation Parameters <= Observations Scale for every factor
Single factor & >= 3 indicators 2 or more factors & 2 or more indicators
per factor
Less than 2 indicators for one or more factors --- ???
Not an issueAs recursive
In literature, 3 indicators or 2 with 2 correlated factors or sample size > 200
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a) How to scale the latent variable 1) fix variances as a constant 2) fix one loading as 1
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b) How to count # parameters = # loadings + vars
& co-vas of factors + vars & co-vas of errors
# obs = v(v+1)/2 ~ number of observed variables
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Examples
A
A
A
X1 X2 X3X2X1
X4X3X1 X2
B
1.0
1.0 1.0
1.0
E1 E2E1
E4
E3E2E1
E2 E3
4, 6, 9
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Identification of EFAGIGO ?
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Estimation Methods ML – most often used
Generalized least squared Un-weighted least squared
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3) Errors and Evaluation: Assumptions
Multivariate normality
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Fit Indices All the fit indices for path analysis applied
to CFA
Chi squared / df < 3 GFI (Goodness Fit Index), AGFI close to 1 SRMR (Standardized Root Mean Squared
Residual) close to 0
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4) Explanation: Factor loadings Un-standardized coefficients (similar to regression coefficients)
Standardized coefficients
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R 2
Proportion of explained variances
(1 – measurement error variance / observed variances)
1-R 2 ~ proportion of unique variances
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Example: The Model Representation
Hand Movements
PhotoSeries
Number Recall
WordOrder
GestaltClosure
Triangles Spatial Memo
MatrixAnalogies
Sequential Simultaneous
1 1
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Example: Results R2
Chi Square Chi-square = 38.13 Df = 19 ~ 2-factor model For one factor 104.90 (df=20)
Indicator R2
Hand .25
Number .65
Word .65
Gestalt .25
Triangle .52
Matrix .43
Spatial .35
Photo .61
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Example: Diagram to Rep Results
Hand Movements
PhotoSeries
Number Recall
WordOrder
GestaltClosure
Triangles Spatial Memo
MatrixAnalogies
Sequential Simultaneous
8.71 (.75)
2.01 (.34)
2.92 (.34)
3.50 (.39)
5.13 (.56)
10.05 (.65)
3.44 (.47)
5.45 (.75)
1.0 (.50)
1.0 (.50)
1.73 (.78)
1.39 (.81)
1.15 (.81)
1.45 (.73)
1.21 (.66)
2.03 (.59)
Standardized coefficients inside parenthesis
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Example: Explanation
Standardized & Un-standardized coefficients & variances (8.71 / 3.4 2 = 8.71 / 11.56 = .75) .5 2 = 1 - .75
Indicator SD
Hand 3.4
Number 2.4
Word 2.9
Gestalt 2.7
Triangle 2.7
Matrix 4.2
Spatial 2.8
Photo 3
Hand Movements
Number Recall
Sequential
8.71 (.75)
2.01 (.34)
1.0 (.50)
1.15 (.81)
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Hybrid Models – Combination of Measurement and Structure Models
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1) Equations and Diagram: Model representation of Hybrid
Model 6 Types of Terms
Observed Exogenous - X Observed Endogenous - Y Latent Exogenous - K Latent Endogenous - E Error Terms for Exogenous Obs – eY
Error Terms for Endogenous Obs - eX
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Diagram representation
K
X
E
Y
eX eY
1, LY2, LX
3, BE4, GA
5, PH 6, PS
7, TE8, TD
eE
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More on Diagram representation
K
X
E1
Y1
eX eY
1, LY2, LX
3, BE4, GA
5, PH
6, PS
7, TE8, TD
E2
Y2 Y3 Y4
eYeYeY
eE1 eE2
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Model Representation NY = # observed endogenous NX = # observed exogenous NE = # latent endogenous NK = # latent exogenous
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Model representation
K
X
E
Y
eX eY
1, LY (NY X NE)2, LX (NX X NK)
3, BE4, GA
5, PH 6, PS
7, TE (NY X NY)8, TD
eE
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2) Identification and Estimation• Number of parameters <(p+q)(p+q+1)/2• Two-Step Rule
- Measurement Model Identification
- Structural Model Identification
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Estimation Methods
ML again
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3) Errors & Model Evaluation Fit Indexes
Chi-squares
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4) Explanation path coefficients
and loadings
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Example: Model
ParentalPsychopathology
Low FamilySES
Reading Arithmetic Spelling Extroversion
FamilialRisk
CognitiveAbility
ScholasticAchievement
Classroom Adjustment
Emotional Stability
MemoryVisual-Spatial
VerbalScholasticMotivation
Harmony
e ee
e e e
ee
e
ee
e
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Example: Identification
ParentalPsychopathology
Low FamilySES
Reading Arithmetic Spelling Extroversion
FamilialRisk
CognitiveAbility
ScholasticAchievement
Classroom Adjustment
Emotional Stability
MemoryVisual-Spatial
VerbalScholasticMotivation
Harmony
e ee
e e e
e
e
e
ee
e
FamilialRisk
CognitiveAbility
ScholasticAchievement
Classroom Adjustment
DD
D
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Example: Errors & Fix Indexes for Evaluation Better chi square/df for 3-factor
measurement model (cognitive & scholar merger) (2.05 vs. 3.92)
(also GFI and SRMR better)
Good chi square/df for hybrid model (2.05)
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Example: results explanation
ParentalPsychopathology
Low FamilySES
Reading Arithmetic Spelling Extroversion
FamilialRisk
CognitiveAbility
ScholasticAchievement
Classroom Adjustment
Emotional Stability
MemoryVisual-Spatial
VerbalScholasticMotivation
Harmony
e ee
e e e
ee
e
ee
e