structural equation modeling

Structural Equation Modeling

Mgmt 291Lecture 3 – CFA and Hybrid Models

Oct. 12, 2009

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.

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)

Beyond Validity and Reliability:Between concept and indicators

Validity: Measures what it intends to measure.

Reliability: Consistency

Precision

repeatability

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

Exploratory Factor Analysis

SPSSFor DataReductionFactor analysis

GIGO

Confirmative Factor Analysis

1) Equations & Diagrams: model representation

2) Identification & Estimation 3) Errors and Evaluation:

assumptions & fit indexes 4) Explanation

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

More on Equations

X = L+ e

Measured

True Score

Error

Relationship between Measured <–> true score Observed <–> latent variable Indicator <–> construct or factor

Uniquefactor

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

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

Non-recursive Type

EducationIncome

Occupation

Socio-economical Status

?

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

a) How to scale the latent variable 1) fix variances as a constant 2) fix one loading as 1

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

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

Identification of EFAGIGO ?

Estimation Methods ML – most often used

Generalized least squared Un-weighted least squared

3) Errors and Evaluation: Assumptions

Multivariate normality

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

4) Explanation: Factor loadings Un-standardized coefficients (similar to regression coefficients)

Standardized coefficients

R 2

Proportion of explained variances

(1 – measurement error variance / observed variances)

1-R 2 ~ proportion of unique variances

Example: The Model Representation

Hand Movements

PhotoSeries

Number Recall

WordOrder

GestaltClosure

Triangles Spatial Memo

MatrixAnalogies

Sequential Simultaneous

1 1

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

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

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)

Hybrid Models – Combination of Measurement and Structure Models

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

Diagram representation

K

X

E

Y

eX eY

1, LY2, LX

3, BE4, GA

5, PH 6, PS

7, TE8, TD

eE

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

Model Representation NY = # observed endogenous NX = # observed exogenous NE = # latent endogenous NK = # latent exogenous

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

2) Identification and Estimation• Number of parameters <(p+q)(p+q+1)/2• Two-Step Rule

- Measurement Model Identification

- Structural Model Identification

Estimation Methods

ML again

3) Errors & Model Evaluation Fit Indexes

Chi-squares

4) Explanation path coefficients

and loadings

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

Example: Identification


Low FamilySES


FamilialRisk

CognitiveAbility



Emotional Stability



Harmony

e ee

e e e

e

e

e

ee

e

FamilialRisk

CognitiveAbility



DD

D

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)

Example: results explanation


Low FamilySES


FamilialRisk

CognitiveAbility



Emotional Stability



Harmony

e ee

e e e

ee

e

ee

e

structural equation modeling

Documents