Causal Models and Structural

Equations Day 1Equations Day 1

Peter Schmidt

Day 1: Overview

1. Course Overview

2. Notation

3. Philosophy of Science and SEM

4. Differences: Exploratory FA and CFA

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4. Differences: Exploratory FA and CFA

5. Unidimensionality

6. Measurement Errors

7. Formative and reflective Indicators

8. Summary and Introduction of practical session

Overview of the Course

1st part: Confirmatory factor analysis

2nd part: Full structural equation model

Course procedure: Regular alternation

3

Course procedure: Regular alternation between

Overview: Types of models 1:

Factor (measurement) Model

A

x1

x

e1

e

4

A x2

x3

e2

e3

Overview: Types of models 1:

Formative Indicator Model

A

x1

xe4

5

A x2

x3

e4

Overview: Types of models 1:

Feedback Model

A

x1

x

e1

ee4

6

A x2

x3

e2

e3

e4

Types of Models in the Course

Factor model (measurement model)- Single or simultaneous analyses of the

measurement models

- Exploratory or confirmatory simultaneous factor analysis

- Multiple group comparison, structured means

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- Multiple group comparison, structured means analyses

- Confirmation, rejection or modification of the models.

- Reflective vs. Formative vs. Feedback indicators

Overview: Types of models 2:

Structural Model

A B

x1y1

e1d1

d3

8

• What are the causal relationships among the

theoretical (latent) variables?

• How strong are these relationships?

• How strong is the stochastic error (d3)?

Ax2

y2 d2e2

Types of Models in the Course

Structural model

- Analysis of the core theory: Is the explication of the

core hypotheses correct?

- MIMIC Model

- Confirmation, rejection or modifications of models

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- Strictly Confirmatory (SC), Alternative Models (AM),

Model Generating (MG)

- Multiple Group Analysis, moderator and non-linear

effects

- Mediators and indirect effects compared to direct

effects

Overview: General Information about the

SEM approach and using AMOS

ADVANTAGES USING SEM

• Test complex hypotheses involving causal relationships among constructs (latent variables).

• Unifies several multivariate methods into one

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• Unifies several multivariate methods into one analytic framework.

• Effects of latent variables on each other and on observed variables.

• Possibility: testing alternative hypotheses.

• Multivariate models without latent variables:

regression models, dummy regressions,

variance analyses and covariance analyses.

• Multivariate models with latent variables:

confirmatory factor analysis (CFA), second order

and nth-order factor analysis, MIMIC models,

canonical correlations, MTMM models, and

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canonical correlations, MTMM models, and

structural equation models (SEM).

• Longitudinal dynamic models: CFA with panel

data, SEM with panel data, autoregressive

models, cross-lagged models, latent growth

curves and differential equations.

Notation: Measurement Model

A

x1

x2

x3

e1

e2

e3

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latent factor (construct)

indicator (observed variable)

measurement error

unidimensionalrelationship

A

x1

e1

Notation: Measurement ModelParameters

A

x1

x2

e1

e2

1.34.74

1 2.13

.58

13

.74 variance of latent construct

1.34 factor loading (unstandardized)

.58 squared multiple correlation

1 Path coefficient of error

2.13 error-variance

x3 e3

Notation: Measurement Model

• correlation, unidimensional path, feedback

foreign antisemitism

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foreign

Correlation

Unidimensional path (effect)

Feedback relation

No Relation!!!

SEM and Philosophy of

Science

• Deductive power

• Transformation of substantive theory

• Operationalizations into confirmatory models

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• Operationalizations into confirmatory models with restrictions to be tested

• Simultaneous test of measurement theory and substantive theory

The methodology provides behavioral

scientists with tools for:

• Stating theories more exactly

• Testing theories more precisely

• Testing alternative theories against each

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• Testing alternative theories against each other

• Generating a more thorough understanding of observed data.

SEM and Philosophy of Science

Lakatos-Kuhn-Scheme:

- metaphysical Assumptions

- Propositions of Core Theory

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- Propositions of Core Theory

- Correspondence Rules

Terminology from Philosophy

of science for theory

construction

Terminology of SEM

Core theory composed of

theoretical postulates (deductive nomological explanation, a�b)

Structural model- causal

relations between constructs

Assumptions of the core theory Assumptions of the structural

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model

Operationalization of theoretical

constructs/dimensions (rules of

correspondence)

Assumptions of

operationalizations (linearity?

Additivity)

Measurement theory- relating

factors to indicators with a set

of assumptions (linearity?

Additivity)

Exploratory Factor Analysis

(orthogonal-no correlation between A1 and A2)

X1=f11A1 + f12A2 + e1

A1 A2f12

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x1 x2 x3 x4

e1 e2 e3 e4

f11

Exploratory Factor Analysis (oblique – factors are correlated)

1212111 ++= δξλξλ1x

1ξ 2ξ12λ

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11λ

1x 2x 3x 4x

1δ 2δ 3δ 4δ

Confirmatory Factor Analysis• X4 = f42A2 + 0*A1 + e4

A1 A2

21

x1 x2 x3 x4

e1 e2 e3 e4

x3

e3

f42

Confirmatory Factor Analysis with correlated factors (CFA) of the theory of planned behavior (with a residual correlation-a non random error)

Pbc

PBC1

PBC2

PBC3

e1

e2

e3

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Subjective

norms

Attitude

NORM1

NORM2

NORM3

Attitu1

Attitu2

Attitu3

e4

e5

e6

e7

e8

e9

Exercise

• Select a theory you are working with

• Select a construct from your theory

• Select some items which measure this construct

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construct

• Draw a measurement model with the respective indicators and constructs

Unidimensionalitiy

• Assumption: A set of Items is explained by only one underlying dimension/construct

x1 e1

24

A

x1

x2

x3

e1

e2

e3

Types of measurement error

• 1) Random measurement error (e‘s): we can

control for it and estimate it if we have at least

two indicators

• 2) Non-random measurement errors (the

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• 2) Non-random measurement errors (the

correlations between the random measurement

errors (e‘s), e.g. social desirability, method

effect): we can control for them and estimate

them if we have at least three indicators, and we

can partly control for them and estimate them

when we have two indicators

Formative and reflective

Indicators

• Reflective: Formative:

A A

eA

26

x1 x2

e1 e2

x1 x2

Summary and Lab Session:

Core theory: Path diagram of the theoretical

assumptions:

Age, gender,

education

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Conformity/Tradition

Allowing immigrants

into the country

Universalism/

Benevolence

Hypotheses:

SH1) The higher the importance of conformity and

tradition, the lower the support for allowing

immigrants into the country.

SH2) The higher the importance of universalism

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SH2) The higher the importance of universalism

and benevolence, the higher the support for

allowing immigrants into the country.

HEitem2e2

11

item1e11

SDitem2e4

item1e311

1

STitem2e6

item1e511

1

UNBE

item2e8

item1e7

11

1

item3e91

item4e101

item5e111

Measurement

model for 7 values

in the ESS

29

COTRitem2e13

item1e12

11

1

SECitem2e17

item1e1611

1

POACitem2e19

item1e18

11

1

item5e11

item3e141

item4e151

item3e201

item4e211

Core theory: Path diagram of the theoretical assumptions (Round 2):

P O A C

im p r i c h

e 1 0

1

1

i p r s p o t

e 1 11

i p s h a b t

e 1 21

i p s u c e s

e 1 31

H E

i p g d t im

e 1 4

1

1

i m p f u n

e 1 51

S T

im p d i f f e 1 611

i p a d v n t e 1 71

i p s t r g ve 2 11

E x 2 : S C F A in t h e N e th e r la n d s , v a lu e s E S S R 2

30

U N B E

i p e q o p t e 11

1

i p u d r s t e 21

i m p e n v e 31

i p h l p p l e 41

T R C O

i p m o d s t

e 6

1

1i m p t r a d

e 7

1i p f r u l e

e 8

1i p b h p r p

e 9

1

S D

ip c r t i v e 1 81 1

im p f r e e e 1 91

S E Cim p s a f ee 2 0

11

i p s t r g ve 2 1

i p l y l f r e 51

An additional research question:

To what extent are the values as proposed to be measured by Shalom Schwartz (1992) equivalent across the three countries Netherlands, Belgium and

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countries Netherlands, Belgium and Luxembourg?

And across a larger set of countries from the ESS?

Summary and Lab Session

Exercise 1: Tradition_conformity in the Netherlands,

ESS R2

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TRCO

ipmodst

e1

1

1

imptrad

e2

1

ipfrule

e3

1

ipbhprp

e4

1

Summary and Lab Session:

The Data

The data we will use in the course:

ESS 2004-2005, focusing on the value questions

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Sample Size:

• The Netherlands: N = 1,881

• Belgium: N = 1,778

• Luxembourg: N = 1,635

• Total sample size: N=5,294

Syntax for generating the

Correlation Matrix

CORRELATIONS

/VARIABLES=selected variables

/PRINT=TWOTAIL SIG

/STATISTICS DESCRIPTIVES

/MISSING=PAIRWISE

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/matrix out (SPSS-file.sav).

���� Example:

CORRELATIONS

/VARIABLES=ipmodst imptrad ipfrule ipbhprp

/PRINT=TWOTAIL SIG

/STATISTICS DESCRIPTIVES

/MISSING=PAIRWISE

/matrix out (cov_nl.sav).