Introduction to Structural Equation Modeling with Amos … · Structural Equation Modeling with Amos ... Confirmatory Factor Analysis ... Factor Analysis: analisis of covariances

Introduction to

Structural Equation Modeling

with Amos

Dr. Lluís Coromina (University of Girona, Spain)

Email: [email protected]

2nd and 3rd October 2014

mailto:[email protected]

1

Outline

Introduction

Basic concepts. Types of variables. Basic composition

Intuitive explanation of the basics of SEM

o Path analysis. The regression analysis model

o Indirect effects. Equations. Degrees of freedom. Specification errors

Measurement errors in regression models

Full SEM model

Confirmatory Factor Analysis (CFA)

o Scale Reliability and Validity of a Construct

SEM and modeling stages.

o Model specification

o Model identification

o Model estimation

o Fit diagnostics and model modification

Results and interpretation

Model modification

2

Introduction

To introduce models that relate variables measured with error.

To introduce Structural Equation Models with latent variables (SEM).

To learn all stages of fitting these models.

To become familiar with the Amos software.

To enable participants to critically read articles in which these models are applied.

3

History

SEM make it possible to:

o Fit linear relationships among a large number of variables. Possibly more than

one is dependent.

o Validate a questionnaire as a measurement instrument.

Quantify measurement error and prevent its biasing effect.

o Freely specify, constrain and test each possible relationship using theoretical

knowledge, testing hypotheses.

In their most recent and advanced versions, SEM enable researchers to:

o Analyze non-normal data.

o Treat missing values by maximum likelihood.

o Treat complex sample data.

4

History of models for the study of causality

Analysis of variance (1920-1930): decomposition of the variance of a dependent variable in order to identify the part contributed by an explanatory variable. Control of third variables (experimental design).

Macroeconometric models (1940-50): dependence analysis of non-experimental data. All variables must be included in the model.

Path analysis (1920-70): analysis of correlations. Otherwise similar to econometric models.

Factor analysis (1900-1970): analysis of correlations among multiple indicators of the same variable. Measurement quality evaluation.

SEM (1970): Econometric models, path analysis and factor analysis are joined together. Relationships among variables measured with error, on non-experimental data from an interdependence analysis perspective.

5

History of models for the study of causality SEM are nowadays very popular because they make it possible to (5 Cs, see Batista &

Coenders 2000):

Work with Constructs/factors/latent variables measured through indicators/observed

variables/manifest variables, and evaluate measurement quality.

Consider the true Complexity of phenomena, thus abandoning uni and bivariate

statistics.

Conjointly consider measurement and prediction, factor and path analysis, and thus

obtain estimates of relationships among variables that are free of measurement error

bias.

Introduce a Confirmatory perspective in statistical modelling. Prior to estimation, the

researcher must specify a model according to theory.

Decompose observed Covariances, and not only variances, from an interdependence

analysis perspective.

6

Basic Concepts

Latent variables (theoretical concepts that cannot be observed directly) =

unobserved = unmeasured

Observed variables (indicators of the underlying construct which they are

presumed to represent)= manifest = measured

7

Basic Concepts

Exogenous (Independent) vs Endogenous (dependent) latent variables.

F1 ‘causes’ F2

Changes in the values of the exogenous variables are not explaine by the model.

Rather, they are considered to be influenced by other factors external to the model

(background variabes such as gender, age, etc.).

Fluctuations in the endogenous variable is said to be explained by the model because

all latent variables that nfluence them are included in the model specification.

8

Statistical Modeling

Models explain how the observed and latent variables are related to one another.

Diagram

Equations

Specification: Model based on researcher’s knowledge of the related theory

Testing on sample data

Goodness of fit between the hypothesized model and sample data. Testing how well

the observed data fit the restricted structure.

Observed data – Hypothesized model = Residual

DATA = MODEL + RESIDUAL

9

Types of variables

Observed variables

Unobserved latent factors

Measurement error associated with an observed variable

Ei =reflects on their adequacy in measuring the related unobserved (underlying) factors.

Residual error (disturbance) in the prediction of an unobserved factor

10

Covariance or correlation

Path coefficient for regression of one factor onto another factor.

Direct relationship

Path coefficient for regression of an observed variable onto an

unobserved latent variable (or factor).

Direct relationship

Spurious relationship: both have a common cause

11

Indirect relationship: both are related by an intervening variable ‘v3’

Joint effect. The difference between ‘Spurious’ and ‘Indirect’. If is that in the latter v1 and v3 are both exogenous so that it is not clear if v3 contributes to the covariance between v1 and v2 through an indirect or spurious mechanism.

12

Factor analytic model

Factor Analysis: analisis of covariances among observed variables in order to get

information of the underlying latent factors.

EFA: Exploratory Factor Analysis. EX: design of a new instrument of measure

‘satisfaction with life’.

CFA: Confirmatory Factor Analysis. Measurement model in Structural Equation

Modeling (SEM). EX: Knowledge of the theory. Hypotesis testing.

Factor loadings: Regression paths from the factors to the observed variables.

13

Full latent variable model

Allows specification of regression structure among latent variables.

Testing of the hypothesis of the impact of one latent construct on another in the

modeling of causal direction.

Full model = measurement model + Structural Model

Recursive Full model: direction of cause from one direction only

Nonrecursive Full model: allows for reciprocal or feedback effects.

14

Example: European Social Survey (ESS)

Estonian data from ESS.

Year: 2012

Ppltrst= Most people can be trusted or you can't be too careful (0= You can't be too

careful; 10= Most people can be trusted)

Pplfair= Most people try to take advantage of you, or try to be fair (0= Most people try

to take advantage of me; 10= Most people try to be fair)

Pplhlp= Most of the time people helpful or mostly looking out for themselves (0=

People mostly look out for themselves; 10=People mostly try to be helpful)

Trstprl= Trust in country's parliament (0= Not trust at all; 10: Complete trust)

Trstplt= Trust in politicians (0= Not trust at all; 10: Complete trust)

Trstlgl= Trust in the legal system (0= Not trust at all; 10: Complete trust)

15

Sample Covariances

ppltrst pplfair pplhlp trstprl trstplt trstlgl

ppltrst 4,956

pplfair 2,698 4,980

pplhlp 2,055 2,225 5,093

trstprl 1,668 1,584 1,369 6,020

trstplt 1,404 1,300 1,103 3,977 5,080

trstlgl 1,705 1,602 1,355 4,153 3,370 6,204

Sample Correlations

ppltrst pplfair pplhlp trstprl trstplt trstlgl

ppltrst 1,000

pplfair ,543 1,000

pplhlp ,409 ,442 1,000

trstprl ,305 ,289 ,247 1,000

trstplt ,280 ,258 ,217 ,719 1,000

trstlgl ,308 ,288 ,241 ,680 ,600 1,000

16

The path model

* Term ei is measurement error (Random measurement error and Systematic error or

non-random)

* Residual (d1) terms represent error in the prediction of endogenous (Political Trust)

factors from exogenous (political Satisfaction) factors.

* All dependent variable have assigned an error (measurement error if it is an observed

variable and a disturbance if it is latent).

17

Basic composition

Measurement model: relations between observed and unobserved variables. CFA:

pattern by which each measure loads on a particular factor.

Structural model: Relations between unobserved variables.

A particular latent variable directly or indirectly influence (‘cause’) changes in the values

of certain other latent variables in the model.

Structural Model Structural Model

Measurement Model

18

Examples and basic concepts. Simple linear regression model.

Introduction to interdependence analysis The specification of a SEM consists in a set of assumptions regarding the behaviour of

the variables involved.

Substantive part: it requires translating verbal theories into equations.

Statistical part: it is needed for the eventual estimation and testing of the model. The

assumptions regard the distribution of the variables involved.

19

Substantive assumptions:

v2=21v1+d2

Linearity.

β21 : effect by how much will the expected value of v2 increase following a unit increase

in v1?

Standardized β21: by how many standard deviations will the expected value of v2

increase following a standard deviation increase in v1?

d2 collects the effect of omitted explanatory variables, measurement error in v2 and the

random and unpredictable part of v2 (disturbance).

v1 is assumed to be free of measurement error.

20

Statistical assumptions regarding the joint distribution of the sources of variation:

22

11

2

1

0

0,

0

0

N

d

v

Two additional parameters: the variances of v1 (11) and d2 (ψ22).

Bivariate normal joint distribution of v1 and d2.

Variables are mean-centred.

Uncorrelation of v1 and d2 (inclusion of all relevant variables).

If this holds, the variance of v2 can be additively decomposed into explained variance

and disturbance variance. R2 is the explained percentage.

Equations exhaustively describe the joint distribution of v1 and v2 as a function of 3

parameters.

21

In order to derive the structural equation system Σ=Σ(π) we can apply path analysis :

2221

1211

For a model with k observed variables, the number of distinct elements in Σ is (k+1)k/2.

Π = (11, ψ22, β21)

2

21112221212222

211121

1111

Determination coefficient R2=1-( ψ22/22)

22

It is possible to solve the system Σ(π)=Σ as it contains an equal number of equations

(distinct elements of Σ) and unknowns (elements of π) exactly identified:

11

2

212222

11

2121

1111

and estimate 212211ˆ,ˆ,ˆ p : by solving

the system Σ(p)=S:

11

2

212222

11

2121

1111

ˆ

ˆ

ˆ

ss

s

ss

s

We can estimate Σ from a sample covariance matrix:

2221

1211

ss

ssS

23

Example

trsrprl =21*ppltrst +d2

(v2=trsrprl) can be explained by level of trust in others (v1=ppltrst):

2221

1211

ss

ssS

(

) = (

)

=

= / = 1.668/4.956= 0.34

=

= 6.020 - (1.6682/4.956)= 5.46

24

21 is identical to the ordinary least squares estimation (dependence analysis).

In statistical analysis, a function of residuals (e.g. the sum of squares) is used as:

A criterion function to minimize during estimation.

A goodness of fit measure.

In a dependence analysis, a residual = 1212ˆ vv .

In an interdependence analysis residuals are differences between covariances fitted by

the model parameters (p) and sample covariances S.

They are arranged in the S-(p) residual matrix.

In an exactly identified model they are zero as S=(p) has a solution.

25

AMOS output for trsrprl =21*ppltrst +d2

Notes for Group (Group number 1)

The model is recursive.

Sample size = 2330

Variable Summary (Group number 1)

Your model contains the following variables (Group number 1)

Observed, endogenous variables

trstprl

Observed, exogenous variables

ppltrst

Unobserved, exogenous variables

d2

Variable counts (Group number 1)

Number of variables in your model: 3

Number of observed variables: 2

Number of unobserved variables: 1

Number of exogenous variables: 2

26

Number of endogenous variables: 1

Parameter Summary (Group number 1)

Weights Covariances Variances Means Intercepts Total

Fixed 1 0 0 0 0 1

Labeled 0 0 0 0 0 0

Unlabeled 1 0 2 0 0 3

Total 2 0 2 0 0 4

Sample Moments (Group number 1)

Sample Covariances (Group number 1)

ppltrst trstprl

ppltrst 4,956

trstprl 1,668 6,020

Sample Correlations (Group number 1)

ppltrst trstprl

ppltrst 1,000

trstprl ,305 1,000

27

Notes for Model (Default model)

Computation of degrees of freedom (Default model)

Number of distinct sample moments: 3

Number of distinct parameters to be estimated: 3

Degrees of freedom (3 - 3): 0

Result (Default model)

Minimum was achieved

Chi-square = ,000

Degrees of freedom = 0

Probability level cannot be computed

28

Estimates (Group number 1 - Default model)

Scalar Estimates (Group number 1 - Default model)

Maximum Likelihood Estimates

Regression Weights: (Group number 1 - Default model)

Standardized Regression Weights: (Group number 1 - Default model)

Estimate

trstprl <--- ppltrst ,305

Variances: (Group number 1 - Default model)

Estimate S.E. C.R. P Label

ppltrst

4,956 ,145 34,125 *** par_2

d2

5,458 ,160 34,125 *** par_3


trstprl <--- ppltrst ,337 ,022 15,479 *** par_1

Critical Ratio= Dividing the regression weight estimate by the estimate of its standard error gives

z = ,337/,022 = 15,479.

Sometimes it is called t-value.

The exogenous variance

(ppltrst) is trivially equal to

the sample variance 4.956.

29

Squared Multiple Correlations: (Group number 1 - Default model)

Estimate

trstprl

,093 ### R2 ###

Standardized Residual Covariances (Group number 1 - Default model)

ppltrst trstprl

ppltrst ,000

In an exactly identified model. trstprl ,000 ,000 they are zero as S=(p) has a solution

30

Simple Regression with SPSS: trsrprl =21*ppltrst +d2

Model R R squared Adjusted R Squared

1 ,305 ,093 ,093

Coeficients

Model Non standardized coefficients

Satandardized coef.

t Sig.

B s.e. Beta

1

(Constant) 2,093 ,129 16,238 ,000

Most people can be trusted or you can't be too careful

,337 ,022 ,305 15,476 ,000

a. Variable dependent: Trust in country's parliament

31

Model with two dependent variables and an indirect effect.

Identification, goodness of fit and specification errors

v2=21v1+d2 v3=32v2+d3

33

22

11

3

2

1

00

00

00

,

0

0

0

N

d

d

v

Σ is 33 and contains 43/2=6 non-duplicated elements.

has 5 elements (11, ψ22, ψ33, β21, β32).

The difference is the number of degrees of freedom (df) of the model.

32

Structural equation system:

32323333

322232

32211131

21212222

211121

1111

EXERCISE: Derive Equation using path analysis.

33

AMOS OUTPUT

Notes for Group (Group number 1)


Sample size = 2330




trstprl

stfeco


ppltrst


d2

d3

34




trstprl

stfeco


ppltrst


d2

d3







35



Fixed 2 0 0 0 0 2

Labeled 0 0 0 0 0 0


Total 4 0 3 0 0 7


ppltrst trstprl stfeco

ppltrst 4,956

trstprl 1,668 6,020

stfeco 1,382 3,007 4,885



ppltrst 1,000

trstprl ,305 1,000

stfeco ,281 ,554 1,000

36








Chi-square = 46,528


Probability level = ,000

Estimates (Group number 1 - Default model) Maximum Likelihood Estimates



trstprl <--- ppltrst ,337 ,022 15,479 *** par_1

stfeco <--- trstprl ,499 ,016 32,156 *** par_2

37


Estimate

trstprl <--- ppltrst ,305

stfeco <--- trstprl ,554



ppltrst

4,956 ,145 34,125 *** par_3

d2

5,458 ,160 34,125 *** par_4

d3

3,383 ,099 34,125 *** par_5

Squared Multiple Correlations: (Group number 1 - Default model) ### R2 ###

Estimate

trstprl

,093

stfeco

,307



ppltrst ,000

trstprl ,000 ,000

stfeco 5,303 ,000 ,000

38

Degrees of freedom introduce restrictions in the covariance space.

From equation:

32323333

322232

32211131

21212222

211121

1111

Implies:

322132211131

22

322131

This derives from many explicit or implicit restrictions of the model.

The existence of degrees of freedom implies higher parsimony. It is a true model in the

scientific sense, which is a simplification of reality.

39

The existence of degrees of freedom affects estimation.

In general, no p vector of estimates will exactly satisfy (p)=S.

Estimation consists in finding a p vector that leads to an S-(p) matrix with small values.

A function of all elements in S-(p)) called fit function is minimized (Fmin)

The existence of degrees of freedom makes it possible to test the model fit. A model

with df=0 leads to a p vector that always fulfils (p)=S or S-(p)=0 and thus perfectly

fits any data set.

In a correct model with df>0 (π)= in the population and (p)S in the sample. If S-

(p), contains large values, we can say that some of the restrictions are false.

If assumptions are fulfilled and under H0 (null hypothesis: the model contains all

necessary parameters), a transformation of the minimum value of the fit function

follows a 2, which makes it possible to test the model restrictions (significance of

omitted parameters). Note that standard testing procedures in statistical modelling

(e.g. t-values) test the parameters which are present in the model.

40

Specification errors

Errors such as the omission of important explanatory variables, the omission of model

parameters, or the inclusion of wrong restrictions are known as specification errors.

Specification errors are frequent. In general, a specification error can bias any

parameter estimate.

If the model is incorrect because v3 receives a direct effect from v1:

v2=21v1+d2

v3=31v1+32v2+d3

and we apply path analysis, then we observe that the new parameter affects σ31 y σ33:

41

313132323333

322232

311132211131

21212222

211121

1111

If we fit the model in to the covariances in the equation, we find σ31 to be affected by

the absent β31 parameter but fitted only by the present parameters β21 and β32. 21 and

32 will be biased.

42

Attempts must be made to detect specification errors by all means, both statistical and

theoretical:

Specification errors are undetectable in any model with df=0. They are also

undetectable if they involve variables that are NOT in the model.

It can happen that many models with different interpretations have a similarly good

fit, even an exactly equal fit (equivalent models).

The following model has a completely different causal interpretation:

v1=12v2+d1 v2=23v3+d2

33

22

11

3

2

1

00

00

00

,

0

0

0

N

v

d

d

EXERCISE: Derive the system for this model and (equivalent to the previous model).

43

If we estimate a general model:

v1=12v2+d1

v2=21v1+23v3+d2

v3=32v2+d3

33

22

11

3

2

1

00

00

00

,

0

0

0

N

d

d

d

then the parameter vector includes 7 elements π=( ψ 11, ψ22, ψ33, β12, β21, β23, β32) versus

6 (3*4/2) equations: infinite number of solutions (underidentified model).

Ψ11

Ψ22 Ψ33

44

Identification of the model

Degrees of freedom (df)= elements matrix S - parameters to be estimated =

[(p)(p+1)]/2 - parameters

Just identified model (df=0): number of data variances and covariances equals the

number of parameters. The model yields an unique solution for all parameters, but

scientifically it is not interesting because without degrees of freedom it never can be

rejected. No goodness of fit of the model is possible.

Over identified model (df>0): It allows to reject the model. It allows to analyze the

discrepancy between S and ∑(p) , thereby rendering it of scientific use. The aim in SEM

then is specify over identified models

Under identified model: infinite number of solutions. No useful at all…

45

Simple regression model with errors in the explanatory variable.

Introduction to models with measurement error The observed explanatory variable (v1) is measured with error (e1). The unobservable

error-free value f1 is called factor or latent variable.

f2 is observed because e2 is for the moment assumed to be zero.

Two equation types:

Relating factors to one another:

f2=β21f1+d2

Relating factors to observed variables or indicators:

v1=f1+e1

v2=f2

Ψ22 θ11 φ11

β21

46

Assumptions:

Measurement errors are uncorrelated with factors (as in factor analysis).

Disturbances are uncorrelated with the explanatory factor (as in regression).

22

11

11

2

1

1

00

00

00

,

0

0

0

N

d

e

f

These assumptions make it possible to decompose the variance of observed variables

into true score variance (explained by factors) and measurement error variance. R2 is

called measurement quality and is represented as .

11

1111

11

11

11

11

2

1 11

47

The structural equations become:

2

21112222

211121

111111

Underidentified model: 4 parameters (11, 11, 21, 22) and three variances and

covariances (only those of observed variables count).

The OLS estimator assumes that 11=0, which is a specification error and leads to bias.

The probability limit of the OLS estimator is:

211

11

211111

11

2111

11

21

11

2121ˆ

s

s

and is thus biased unless κ1=1.

48

Simple linear regression model with multiple indicators The solution to measurement error bias in SEM involves the use of multiple indicators,

at least of the explanatory latent variables.

The equations relating factors to indicators become:

f2=β21f1+d2

v1=1* f1+e1

v2=f2

v3=λ31*f1+e3

49

The equation includes a loading λ31 (L31) which relates the scales of f1 and v3:

The researcher must fix the latent variable scale, usually by anchoring it to the

measurement units of an indicator whose equals 1.

Standardized instead of raw loadings are usually interpreted. If there is only one

factor per indicator, they lie within -1 and +1 and equal the square root of κ.

New assumption of uncorrelated measurement errors of different indicators:

22

33

11

11

2

3

1

1

000

000

000

000

,

0

0

0

0

N

d

e

e

f

50

Addition of v3 in the structural equations. This is an exactly identified model, all of

whose parameters can be solved, even those related to unobservable variables. The

extent to which multiple indicators of the same construct converge (correlate) provides

information to estimate the parameters.

2

21112222

2

31113333

111111

112121

113131

32312111

33

2

311133

31211132

311131

2

21112222

211121

1111

2

11

1

1

1

1

51

Applied example f2=v2=trstprl from Social Trust f1=SocT, measured by its two indicators (v1=ppltrst and v3=pplhlp):

ppltrst = 1* SocT+ e1

pplhlp = 31* SocT+e3

trstprl=trstprl

trstprl=β21 * SocT+d2

52

The equivalence between the observed and latent dependent variable makes it possible

to simplify the path and equations as:

ppltrst = 1* SocT+ e1

pplhlp = 31* SocT+e3

trstprl= β21 * SocT+d2

We define a latent variable called Social Trust, measured by ppltrst and pplhlp. The

loading of ppltrst (first indicator) is constrained to 1 in order to fix the scale of the latent

variable. Each indicator automatically receives a (error variance, ei) parameter.

The regression is of trstprl (observed, dependent) on Social Trust (latent, explanatory).

This automatically defines a (regression weight), a (variance of independent

variable) and a (disturbance term) parameter.

θ11

θ33

ψ22

φ1

1

λ31

β21

53

AMOS OUTPUT

54


Sample size = 2330




ppltrst

pplhlp

trstprl


SocialTrust

e3

e1

d2







55



Fixed 4 0 0 0 0 4

Labeled 0 0 0 0 0 0


Total 6 0 4 0 0 10



trstprl pplhlp ppltrst

trstprl 6,020

pplhlp 1,369 5,093

ppltrst 1,668 2,055 4,956


trstprl pplhlp ppltrst

trstprl 1,000

pplhlp ,247 1,000

ppltrst ,305 ,409 1,000

56








Chi-square = ,000


Probability level cannot be computed

57


Scalar Estimates (Group number 1 - Default model) Maximum Likelihood Estimates



ppltrst <--- SocialTrust 1,000

fixed at 1

pplhlp <--- SocialTrust ,821 ,069 11,972 *** par_1 free 31

trstprl <--- SocialTrust ,666 ,056 11,954 *** par_2

Standardized Regression Weights:

Estimate

ppltrst <--- SocialTrust ,711

pplhlp <--- SocialTrust ,575

trstprl <--- SocialTrust ,430


Estimate We can compute R2

trstprl

,185 0,4752

= 0,185

pplhlp

,331 0,5752

= 0,331

ppltrst

,505 0,7112 = 0.505

58



SocialTrust

2,505 ,239 10,496 *** par_3 11

e3

3,407 ,169 20,161 *** par_4 33

e1

2,452 ,215 11,410 *** par_5 11

d2

4,909 ,170 28,942 *** par_6 22

Matrices (Group number 1 - Default model)

Implied (for all variables) Covariances (Group number 1 - Default model)

SocialTrust trstprl pplhlp ppltrst

SocialTrust 2,505

trstprl 1,668 6,020

pplhlp 2,055 1,369 5,093

ppltrst 2,505 1,668 2,055 4,956

Implied (for all variables) Correlations (Group number 1 - Default model)

SocialTrust trstprl pplhlp ppltrst

SocialTrust 1,000

trstprl ,430 1,000

pplhlp ,575 ,247 1,000

ppltrst ,711 ,305 ,409 1,000

59

Full SEM model

Example:

Independent variables (8):

6 errors: e1, e2, e3, e4, e5, e6

1 disturbance: d1

1 latent variable: SocialTrust

Dependent variables (7):

6 represent observed variables: ppltrst; pplfair; pplhlp; trstprl; trstplt; trstlgl

1 represents an unobserved variable (or factor Political Trust).

60

Definition of the model:

Model Equations

Political Trust = ? * SocialTrust + d1 Structural model

ppltrst = 1* SocialTrust + e1

pplfair = ? * SocialTrust + e2 Social Trust

pplhlp = ? * SocialTrust + e3 measurement scale model

trstprl = 1*Political Trust + e4

trstplt = ? * Political Trust + e5 Political trust

trstlgl =? * Political Trust + e6 measurement scale model

Variances of independent variables

e1 = ? ; e2 = ?; e3 = ?; e4=?; e5 = ?; e6 = ?; SocialTrust = ?; d1 = ?

61

Rules for determining the model parameters

• Rule 1: All the variances of the independent variables are parameters

• Rule 2: All covariances between independent variables are parameters

• Rule 3: All load factors between latent and its indicators are parameters

• Rule 4: All regression coefficients between observed or latent variables are

parameters

• Rule 5: (i) The variances of dependent variables, (ii) the covariance between

dependent variables and (iii) the covariance between dependent and independent

variables, are never parameters (are explained by other parameters of the model)

• Rule 6: For each latent variable must be set its metric:

– For independent latent two ways:

• Set its variance set to a constant (usually 1)

• Fix a load factor (λ) between latent and its factor (usually 1)

62

Determining the model parameters:

For the latent dependent only one way: fix a coefficient between it and one of the

observed variables to a constant (usually 1)

• An equation for each variable (latent or observable) that receives a one-way arrow

(dependent variables) (7)

• So many variances as independent variables (8)

• So many covariances as two-way arrows [0]

63


Number of distinct sample moments: 21 = = (6 * 7/2)


8 variances of independent variables

4 coefficients of latent factors with indicators

1 regression coefficient


64

Determining the model parameters:

Adding a covariance between e5 and e6, we introduce a parameter, and we lose one

degree of freedom.





65

Confirmatory Factor Analysis CFA.

Introduction to reliability and validity assessment

φ12

λ11

λ21

λ32

λ42

θ11

2

θ22

θ44

θ33

66

This model does not contain equations relating factors to one another but only

covariances. All factors are exogenous. No or parameters, only , θ and φ.

At least three indicators are needed for models with one factor and two for models

with more factors.

In CFA models it is possible to standardize factors to unit variances instead of fixing a

loading to 1. Then the parameters are factor correlations.

For 2 factors and 2 indicators we have the following equations:

v1=11f1+e1 v2=21f1+e2

v3=32f2+e3 v4=λ42f2+e4

67

The model has df=1. 11=22=1:

44

33

22

11

21

21

4

3

2

1

2

1

00000

00000

00000

00000

00001

00001

,

0

0

0

0

0

0

N

e

e

e

e

f

f

44

2

4244

324243

21422142

11422141

33

2

3233

21322132

11322131

22

2

2122

211121

11

2

1111

1

1

1

1

1

1

68

The correlation between two indicators of the same factor depends on :

11

1111

11

11

11

11

2

1 11

21

22

2

2111

2

11

2

21

2

11

22

2

2111

2

11

2111

2211

2121

and the correlation between two indicators of different factors is attenuated with

respect to the correlation between factors (effect of measurement error):

3121

33

2

3211

2

11

2

32

2

1121

33

2

3211

2

11

322111

3311

3131

A CFA model is likely to fit the data only if items of the same factor correlate highly and

higher than items of different factors. We advise researchers to carefully examine the

correlation matrix prior to fitting a CFA model.

69

Reliability for each item:

Trstprl κ1=

=0.735 trstlgl κ2=

=0.629

stfgov κ3=

=0.789 stfdem κ4= ???

Correlations:

2121 =√ =0.680

312131 =0.862*√ =0.669

41 ????

32 ????

34 ????

42 ????

70

Amos Output




trstlgl

trstprl

stfdem

stfgov


e2

e1

e4

SATCNTRY

e3

PolTrust







71


stfgov stfdem trstprl trstlgl

stfgov 5,534

stfdem 3,792 5,182

trstprl 3,861 3,152 6,020

trstlgl 3,446 3,304 4,153 6,204


stfgov stfdem trstprl trstlgl

stfgov 1,000

stfdem ,708 1,000

trstprl ,669 ,564 1,000

trstlgl ,588 ,583 ,680 1,000






72


Scalar Estimates (Group number 1 - Default model)




stfgov <--- SATCNTRY 2,089 ,042 49,720 *** par_1

stfdem <--- SATCNTRY 1,815 ,042 43,216 *** par_3

trstlgl <--- PolTrust 1,976 ,046 42,608 *** par_4

trstprl <--- PolTrust 2,102 ,045 46,963 *** par_5


Estimate λi

stfgov <--- SATCNTRY ,888

stfdem <--- SATCNTRY ,797

trstlgl <--- PolTrust ,793

trstprl <--- PolTrust ,857

73

Squared Multiple Correlations: 1-λi2=θi

Estimate

stfgov

0,789 1-0,789=0,211

stfdem

0,635 1-0,635=0,365

trstlgl

0,629 1-0,629=0,371

trstprl 0,735 1-0,735=0,265

Covariances: (Group number 1 - Default model)


SATCNTRY <--> PolTrust ,862 ,011 76,185 *** par_2

Correlations: (Group number 1 - Default model)

Estimate

SATCNTRY <--> PolTrust ,862 The same value than covariance. This is because variance of latent

factors is fixed to 1. 11=22=1

74



PolTrust

1,000

SATCNTRY

1,000

e2

2,300 ,098 23,458 *** par_6

e1

1,601 ,093 17,164 *** par_7

e4

1,887 ,079 23,765 *** par_8

e3

1,169 ,083 14,103 *** par_9


PolTrust SATCNTRY stfgov stfdem trstprl trstlgl

PolTrust 1,000

SATCNTRY ,862 1,000

stfgov ,766 ,888 1,000

stfdem ,688 ,797 ,708 1,000

trstprl ,857 ,739 ,656 ,589 1,000

trstlgl ,793 ,684 ,608 ,545 ,680 1,000

75

Purification of the measures

Total item correlation serves as a criterion for initial assessment and purification.

Various cut-off points are adopted:

0.30 by Cristobal et al.(2007)

0.40 by Loiacono et al. (2002)

0.50 by Francis and White(2002) and Kim and Stoel (2004)

Wolfinberger and Gilly (2003) are rigorous in retaining only items that

load at 0.50 or more on a factor

do not load at more than 0.50 on two factors

have an item total correlation of more than 0.40

76

Reliability and Validity

A measure is reliable to the extent that independent but comparable measures of the

same trait or construct of a given object agree. Reliability depends on how much of

the variation in scores is attributable to random or chance errors. If a measure is

perfectly reliable, XR = 0

A measure is valid if when the differences in observed scores reflect true differences

on the characteristic one is attempting to measure and nothing else, that is, XO= XT.

a) Reliability of individual items:

o loadings greater than 0.50 on the respective construct (Hulland, 1999; White

et al., 2003; Ribbink et al. 2004)

o exhibit loadings with the intended construct of .70 or more, and are

statistically significant (Ledden, 2007)

77

b) Reliability of a construct or Internal consistency of the scale

It allows to check the internal consistency of all indicators to measure the concept

(thoroughness with which all indicators measure the same)

…Internal homogeneity of a set of items:

o Composed Reliability (CR) greater than 0.70 (Anderson and Gerbing, 1988;

Bagozzi and Yi, 1988)

o Correlation between each item and its construct >0.5 and correlations among

items from the same construct >0.3.

Cronbach’s α greater than 0.70 (Fornell and Larcker, 1981; Nunally and

Bernstein, 1994)

78

c) Convergent validity

…the extent to which a set of items assumed to represent a construct does in fact

converge on the same construct:

o Average variance extracted (AVE - amount of variance that a construct obtains

from the indicators in relation to the amount of variance of the measurement

error) greater than 0.50 (Fornell and Larcker, 1981; Chin and Newsted, 1999;

Gounaris and Dimitriadis, 2003)

o Factor loadings greater than 0.5 (Grewal et al., 1998).

• (d) Discriminant validity (when there are some scales in the model)

…the extent to which measures of theoretically unrelated constructs do not correlate

with one another:

– inter-factor correlations are less than the square root of the average variance

extracted (AVE) (Fornell and Larcker, 1981)

79

Numerical example:

Reliability of a construct

Reliability (SATCNTRY)=

=0.831

Reliability (PolTrust)=

=0.811

80

Convergent validity

1st) Factor loadings are significant and greater than 0.5

2nd) Average Variance Extracted (AVE) for each of the factors > 0.5.

Squared Multiple Correlations: AVE

Estimate

stfgov

0,789 (0.789+0.635)/2= 0.712 SATCNTRY

stfdem

0,635

trstlgl

0,629 (0.629+0.735)/2= 0.682 PolTrust

trstprl 0,735

81

Discriminant Validity of construct

Average variance extracted (AVE). For this a construct must have more variance with its

indicators than with other constructs of the model. It is when

√ between each pair of factors > estimated correlation between those factors

StfCntry PolTrust

StfCntry 0.843

PolTrust 0.862 0.823

sqrt(0.712)=0.832 sqrt(0.682)=0.823

Correlations:

Estimate

SATCNTRY <--> PolTrust ,862

82

Exploratory Factor Analysis (EFA)

83

Comparison with exploratory factor analysis (EFA)

v1=11f1+12f2+e1

v2=21f1+22f2+e2

v3=31f1+32f2+e3

v4=λ41f1+42f2+e4

44

33

22

11

4

3

2

1

2

1

00000

00000

00000

00000

000010

000001

,

0

0

0

0

0

0

N

e

e

e

e

f

f

In EFA, which items measure which dimensions is the outcome, in CFA it is the input.

In EFA questionnaire items aim globally at a broad concept, in CFA each questionnaire

item is designed to tackle a specific dimension of the concept.

84

Random and systematic error. Reliability and validity assessment with

CFA Reliability: Extent to which a measurement procedure “would” yield the same result upon several independent trials under identical conditions. In other words, low random measurement error (any systematic error would replicate). Random measurement error is a problem for OLS regression but not for SEM with multiple indicators, because it is

accounted for by the parameters.

Validity: Extent to which a measurement procedure measures what it is intended to measure and only what it is intended to measure, except for random measurement error. In other words, absence of systematic error.

Assuming the validity of v, its reliability is the percentage of variance explained by f.

Always follow this golden rule:

Estimate reliability after validity has been diagnosed.

Test the specification of measurement equations in a CFA model prior to specifying equations relating factors. Otherwise, relationships among factors might be biased (specification errors) or even meaningless (invalidity).

85

Construct validation: Estimate a CFA model that assumes validity...

All items load on the factor they are supposed to measure (a second loading is a sign of measuring another factor which is in the model).

No error correlations are specified (error correlations contain common unknown variance, a sign of measuring an unknown factor which is not in the model).

....and diagnose its goodness of fit. You can never be certain of validity, but a CFA model can help detect signs of invalidity such as:

It does not correctly reproduce the covariance matrix (additional loadings or error correlations are needed, thus revealing mixed items, additional necessary dimensions). Convergent invalidity.

Some variables have too low to be attributed to solely random error (convergent invalidity).

Some factors have correlations very close to unity (discriminant invalidity). Some factors have correlations of unexpected signs or magnitudes (nomological

invalidity).

86

Modelling stages in SEM

1) SPECIFICATION

2) IDENTIFICATION

3) DATA COLLECTION

4) ESTIMATION

5) FIT DIAGNOSTICS

ADEQUATE?

6) UTILIZATION

YES

NO

Model: equations and assumptions

Estimable model

Exploratory data analysis. Computation of S

Methods to fit Σ(p) to S

Discrepancies between Σ(p) and S

Verbal theories

- Theory validation, validity and reliability assessment ...

MODIFICATION

87

Theoretical and statistical grounds

1) Specification

Formal establishment of a statistical model: set of statistical and substantive assumptions that structure the data according to a theory.

Equations: one or two of the following systems of equations: Relating factors or error free variables to one another (structural equations). Relating factors to indicators with error (measurement equations).

Parameters: two types: o Free (unknown and freely estimated). o Fixed (known and constrained to a given value, usually 0 or 1).

The amount of the researchers’ prior knowledge will affect the modelling strategy: If this knowledge is exhaustive and detailed, it will be easily translated into a model specification. The researchers’ aim will simply be to use the data to estimate and confirm or reject the model (confirmatory strategy). If this knowledge is less exhaustive and detailed, the fixed or free character of a number of parameters will be dubious. This will lead to a model modification process by repeatedly going through the modelling stages (exploratory strategy).

88

Full SEM model:

89

Equations: StfCntry = 31 * SocialTrust + d3 PolTrust = 21 * SocialTrust + d2

ppltrst = 1* SocialTrust + e1 pplfair = 21 * SocialTrust + e2 pplhlp = 31 * SocialTrust + e3

trstprl = 1*Political Trust + e4 trstplt = 52 * Political Trust + e5 trstlgl =62 * Political Trust + e6

stfeco=73 * StfCntry + e7 stfgov=83 * StfCntry + e8 stfdem=1 * StfCntry + e9 Variances of independent variables e1 = 11 ; e2 = 22; e3 = 33; e4=44; e5 = 55; e6 = 66; e7 = 77; ; e8 = 88; ; e9 = 99; SocialTrust = φ11; d2 = ψ22; d3 = ψ33

Total number of

parameters = 20

90

2) Identification Can model parameters be derived from variances and covariances?

Identification must be studied prior to data collection

If a model is not identified: o Seek more restrictive specifications with additional constraints (if theoretically

justifiable). o Add more indicators or more exogenous factors.

Identification conditions

o Underidentification (df<0): infinite number of solutions that make S equal (p). o Possibly identified (df=0): there may be a unique solution that makes S equal

(p). This type of models is less interesting in that their restrictions are not testable.

o Possibly overidentified (df>0): there may be a unique solution that minimizes

discrepancies between S and (p). Only these models, more precisely their restrictions, can be tested from the data.

91

Example Full SEM model:

9 observed variables lead to (910/2)=45 variances and covariances: possibly overidentified model.

Total number of parameters = 20

Degrees of Freedom = 45-20 = 25 > 0

The model fulfils enough sufficient conditions:

1) Equations relating factors are recursive

2) Disturbances are uncorrelated

3) All factors have at least two pure indicators.

92

3) Data collection and exploratory analyses

Valid sampling methods

In their standard form, SEM assumes simple random sampling. Extensions to stratified and cluster samples have been recently developed. In any case, they must be random samples.

Sample size

Sample sizes in the 200-500 range are usually enough. Sample requirements increase:

For smaller R2 and percentages of explained variance.

When collinearity is greater.

For smaller numbers of indicators per factor (especially less than 3).

Under non normality, the required sample size is larger (in the 400-800 range).

Outlier and non-linearity detection

As before doing any other type of statistical modeling, outliers and non-linear relationships must be detected by means of exploratory data analysis.

93

4) Estimation

First estimate the sample variances and covariances (S) and then find the best fitting

p parameter values.

A fit function related to the size of the residuals in S-(p) is minimized.

Each choice of fit function results in an alternative estimation method. One of

these choices leads to the maximum likelihood estimator (ML) which is the most

often used.

Estimation assumes that a covariance matrix is analyzed. Estimations obtained

from a correlation matrix are only correct only under very specific conditions.

94

Normality assumed: ML and GLS

Normality not assumed: ULS, Scale LS, AD

The two most commonly used estimation techniques are Maximum likelihood (ML) and normal theory generalized least square (GLS).

ML and GLS: large sample size, continuous data, and assumption of multivariate normality

Unweighted least squares (ULS): scale dependent.

Asymptotically distribution free (ADF) (Weighted least squares, WLS): serious departure from normality.

95

Examining the ordered correlation matrix

Let us look at correlations as well and spot low correlations of items measuring the

same or large correlations between items measuring different dimensions:

stfeco stfgov stfdem ppltrst pplfair pplhlp trstprl trstplt trstlgl

stfeco 1,000

stfgov ,718 1,000

stfdem ,653 ,708 1,000

ppltrst ,281 ,276 ,276 1,000

pplfair ,285 ,273 ,275 ,543 1,000

pplhlp ,298 ,277 ,262 ,409 ,442 1,000

trstprl ,554 ,669 ,564 ,305 ,289 ,247 1,000

trstplt ,515 ,642 ,530 ,280 ,258 ,217 ,719 1,000

trstlgl ,526 ,588 ,583 ,308 ,288 ,241 ,680 ,600 1,000

96

Amos output




trstlgl

trstplt

trstprl

pplhlp

pplfair

ppltrst

stfdem

stfgov

stfeco

Unobserved, endogenous variables

PolTrust

StfCntry


e6

e5

e4

SocialTrust

e3

e2

e1

e9

e8

e7

d3

d2

97














Chi-square = 1097,457



98





PolTrust <--- SocialTrust 1,985 ,101 19,721 *** par_7

StfCntry <--- SocialTrust 1,670 ,087 19,200 *** par_8

trstlgl <--- PolTrust ,897 ,021 43,529 *** par_1

trstplt <--- PolTrust ,845 ,018 46,008 *** par_2

trstprl <--- PolTrust 1,000

pplhlp <--- SocialTrust ,911 ,064 14,299 *** par_3

pplfair <--- SocialTrust ,984 ,065 15,135 *** par_4

ppltrst <--- SocialTrust 1,000

stfdem <--- StfCntry 1,000

stfgov <--- StfCntry 1,155 ,024 47,207 *** par_5

stfeco <--- StfCntry ,975 ,023 42,054 *** par_6

99


Estimate

PolTrust <--- SocialTrust ,904

StfCntry <--- SocialTrust ,899


trstplt <--- PolTrust ,806


pplhlp <--- SocialTrust ,396

pplfair <--- SocialTrust ,432

ppltrst <--- SocialTrust ,440

stfdem <--- StfCntry ,800

stfgov <--- StfCntry ,894

stfeco <--- StfCntry ,803

Variances:


SocialTrust

,960 ,093 10,320 *** par_9

d3

,636 ,076 8,417 *** par_10

d2

,841 ,106 7,927 *** par_11

e6

2,478 ,090 27,622 *** par_12

e5

1,778 ,068 25,983 *** par_13

e4

1,393 ,071 19,718 *** par_14

e3

4,296 ,130 32,982 *** par_15

e2

4,050 ,124 32,705 *** par_16

e1

3,996 ,122 32,639 *** par_17

e9

1,868 ,069 26,999 *** par_18

e8

1,115 ,060 18,450 *** par_19

e7

1,737 ,065 26,838 *** par_20

100

Squared Multiple Correlations:

Estimate

StfCntry

,808

PolTrust

,818

stfeco

,644

stfgov

,799

stfdem

,640

ppltrst

,194

pplfair

,187

pplhlp

,156

trstprl

,769

trstplt

,650

trstlgl

,601

101


SocialTrust StfCntry PolTrust stfeco stfgov stfdem ppltrst pplfair pplhlp trstprl trstplt trstlgl

SocialTrust 1,000

StfCntry ,899 1,000

PolTrust ,904 ,813 1,000

stfeco ,722 ,803 ,653 1,000

stfgov ,803 ,894 ,727 ,717 1,000

stfdem ,719 ,800 ,650 ,642 ,715 1,000

ppltrst ,440 ,396 ,398 ,318 ,354 ,316 1,000

pplfair ,432 ,389 ,391 ,312 ,347 ,311 ,190 1,000

pplhlp ,396 ,356 ,358 ,285 ,318 ,284 ,174 ,171 1,000

trstprl ,793 ,713 ,877 ,572 ,637 ,570 ,349 ,343 ,314 1,000

trstplt ,729 ,655 ,806 ,526 ,586 ,524 ,321 ,315 ,288 ,707 1,000

trstlgl ,701 ,630 ,775 ,506 ,563 ,504 ,309 ,303 ,277 ,679 ,625 1,000

102

5) Fit diagnostics

Interpretation does not proceed until the goodness of fit has been assessed.

The fit diagnostics attempt to determine if the model is correct and useful. o Correct model: its restrictions are true in the population. Relationships are

correctly specified without the omission of relevant parameters.

o In a correct model, the differences between S and (p) are small and random. o Correctness must not be strictly understood. A model must be an approximation

of reality, not an exact copy of it. o Thus, a good model will be a compromise between parsimony and

approximation.

Diagnostics will usually do well at distinguishing really badly fitting models from fairly well fitting models. Many models will fit fairly well (even exactly equally well if equivalent) and will be hard to distinguish statistically, they can be only distinguished theoretically.

103

The 2 goodness of fit statistic

Null hypothesis: the model is correct, without omitted relevant parameters:

H0:=()

2 goodness of fit statistic follows a 2 distribution with g degrees of freedom. Rejection implies concluding that some relevant parameters have been omitted.

Sample size and power of the test are often high. Researchers are usually willing to accept approximately correct models with small misspecifications, which are rejected due to the high power. Quantifying the degree of misfit is more useful than testing the hypothesis of exact fit.

104

5.1 Global diagnostics

First look for serious problems (common for small samples, very badly fitting models, and models with two indicators per factor): o Lack of convergence of the estimation algorithm. o Underidentification.





Degrees of freedom (10 - 12): -2


The model is probably unidentified. In order to achieve identifiability, it will probably be necessary to impose 1 additional

constraint.

If Inadmissible estimates (e.g. negative variances, correlations larger than 1...).

105

Fix negative unsignificant variances to zero:

Revise the model if there are significant negative variances.

Merge in one pairs of factors with correlations larger than 1 or not significantly lower

than 1.

106

The Tucker and Lewis’ (1973) index (TLI) and Bentler’s (1990) comparative fit index

(CFI) introduce the degrees of freedom of the base (gb) and researcher (g) models to

account for parsimony. They will increase after adding parameters only if the 2

statistic decreases more substantially than g.

12

22

b

b

b

b

g

ggTLI

1;

)()(min

2

22

bb

bb

g

ggCFI

Root mean squared error of approximation (RMSEA) (Steiger, 1990):

gN

gRMSEA

0;max 2

Values below 0.05 are considered acceptable.

The sampling distribution is known, which makes it possible to do confidence

intervals and test the hypothesis of approximate fit. If both extremes of the interval

are larger than 0.05, a very bad fit can be concluded. If both extremes are below 0.05,

a very good fit can be concluded.

107

5.2. Detailed diagnostics

Are standardized estimated values reasonable and of the expected sign?

Are there significant residuals that suggest the addition of parameters? (To estimate them in Amos Analysis properties \ Output \ Residual moments). The values are t-values.

Each residual covariance, has been divided by an estimate of its standard error. In sufficiently large

samples, these standardized residual covariances have a standard normal distribution if the model is

correct. So, if the model is correct, most of them should be less than two in absolute value.

Are there low R2 values suggesting the omission of explanatory variables or low values suggesting a lack of validity? (To estimate them in Amos Analysis properties \ Output \ Squared Multiple Correlations)

108

The modification index is an individual significance test of omitted parameters (Ho: the omitted parameter is zero in the population).

o Reject hypothesis above critical 2 value with 1 df. 3.84 for type I risk 5%. o Always consider the expected standardized estimated parameter and its sign: if

power is high, parameters of a substantially insignificant value can be statistically significant. Only add parameters of a substantial size.

Residuals and modification indices can suggest the addition of parameters in order to improve fit. A model can also be improved by dropping irrelevant parameters (parsimony principle). The usual t statistic tests the significance of included parameters (Ho: the included parameter is zero in the population).

o Non-significant disturbance covariances and measurement error covariances

should be dropped from the model. Non-significant parameters may be dropped

from the model if their theoretical argumentation is weak. Non -significant parameters reveal invalidity.

109


stfeco stfgov stfdem ppltrst pplfair pplhlp trstprl trstplt trstlgl

stfeco ,000

stfgov ,037 ,000

stfdem ,442 -,255 ,000

ppltrst -1,694 -3,550 -1,881 ,000

pplfair -1,257 -3,363 -1,656 16,729 ,000

pplhlp ,569 -1,851 -1,039 11,168 12,884 ,000

trstprl -,739 1,307 -,236 -1,988 -2,438 -3,059 ,000

trstplt -,483 2,350 ,262 -1,888 -2,613 -3,316 ,487 ,000

trstlgl ,888 1,053 3,394 -,047 -,685 -1,682 ,010 -1,004 ,000

110

Modification Indices


M.I. Par Change

d2 <--> d3 37,757 ,224

e7 <--> d2 10,661 -,152

e8 <--> d2 59,293 ,325

e1 <--> d3 68,710 -,454

e1 <--> d2 21,676 -,304

e1 <--> e8 44,484 -,380

e2 <--> d3 55,293 -,410

e2 <--> d2 40,987 -,421

e2 <--> e8 42,665 -,374

e2 <--> e1 474,885 1,882

e3 <--> d3 9,736 -,177

e3 <--> d2 71,245 -,570

e3 <--> e7 7,221 ,170

111

Model Fit Summary

RMR, GFI

Model RMR GFI AGFI PGFI

Default model ,417 ,894 ,809 ,496

Saturated model ,000 1,000

Independence model 2,251 ,366 ,207 ,293

Baseline Comparisons

Model NFI

Delta1

RFI

rho1

IFI

Delta2

TLI

rho2 CFI

Default model ,897 ,852 ,899 ,854 ,899

Saturated model 1,000

1,000

1,000

Independence model ,000 ,000 ,000 ,000 ,000

RMSEA

Model RMSEA LO 90 HI 90 PCLOSE

Default model ,136 ,129 ,143 ,000

Independence model ,356 ,350 ,361 ,000

112

5.3. Model modification. Capitalization on chance

Frequently models fail to pass the diagnostics.

Which modifications introduce and in which order?

o Introduce modifications one at a time, and carefully examine results before introducing the next. One modification can modify the need for another.

o First improve fit (add parameters). Then improve parsimony (drop parameters).

o Disregard high modification indices with very small expected estimates.

o Consider models with good descriptive fit indices, even if the 2 test rejects them (parsimony-approximation compromise).

o Avoid adding theoretically uninterpretable parameters, no matter how significant.

o Make few modifications.

o The selected model must pass the diagnostics, theoretically relevant and useful.

o Modified models can be compared with CFI and RMSEA.

113

Model modification has some undesirable statistical consequences, especially if

modifications are blindly done using only statistics, that is, without theory.

Even if model modification has been done carefully, modifications are based on a

particular sample. Have we reached a model that fits the population?

Bias of estimates and significance tests: only large and significant parameters have

been considered to be candidates for addition.

The introduction of modifications that improve the fit to the sample but not to the

population is known as capitalization on chance.

The only solution is to check that the model fits well beyond the particular sample

used:

o Crossvalidation: estimation and goodness of fit test of the model on an independent sample of the same population. If only one sample is available, it can be split: the first half is used for model modification and the second for validation. Crossvalidation is successful if the model fits the second sample reasonably well.

114

Complete Example Modeling Stages:

CFA model

115

trstprl = 1*Political Trust + e4

trstplt = 52 * Political Trust + e5 trstlgl =62 * Political Trust + e6

stfeco=73 * StfCntry + e7

stfgov=83 * StfCntry + e8

stfdem=1 * StfCntry + e9

Variances of independent variables

e4=44; e5 = 55; e6 = 66; e7 = 77; ; e8 = 88; ; e9 = 99;

Cov(StfCntry,Poltrust)= φ23; Var(StfCntry) = φ33;

Var(polTrust) = φ22;

116

The model has 6x7/2=21 variances and covariances, and 13 parameters (6 error

variances , 2 factor variances, 1 factor covariance, 4 loadings): 8 degrees of freedom.

Each factor has at least 2 pure indicators: the measurement part is identified.

In the complete model with parameters, the factors are related in a recursive system

without error covariances: it is identified.

This model only has measurement equations. The loading of the first variable in each

factor is equal to 1. The remaining loadings are free. Each observed variable also has

an error variance . By default all factor variances and covariances are free. To

constrain factors to be uncorrelated one would add the constrained parameter with a

value of 0 in the Covariance:

117

Estimation



trstlgl

trstplt

trstprl

stfdem

stfgov

stfeco


PolTrust

e6

e5

e4

StfCntry

e9

e8

e7







118



Fixed 8 0 0 0 0 8

Labeled 0 0 0 0 0 0


Total 12 1 8 0 0 21



stfeco stfgov stfdem trstprl trstplt trstlgl

stfeco 4,885

stfgov 3,734 5,534

stfdem 3,284 3,792 5,182

trstprl 3,007 3,861 3,152 6,020

trstplt 2,565 3,405 2,721 3,977 5,080

trstlgl 2,898 3,446 3,304 4,153 3,370 6,204

119



stfeco 1,000

stfgov ,718 1,000

stfdem ,653 ,708 1,000

trstprl ,554 ,669 ,564 1,000

trstplt ,515 ,642 ,530 ,719 1,000

trstlgl ,526 ,588 ,583 ,680 ,600 1,000








Chi-square = 121,665



120

Estimates (Group number 1 - Default model). Maximum Likelihood Estimates



trstlgl <--- PolTrust ,893 ,021 43,329 *** par_1 ### Significant###

trstplt <--- PolTrust ,848 ,018 46,413 *** par_2 ### Significant###



stfgov <--- StfCntry 1,174 ,025 47,430 *** par_3 ### Significant###

stfeco <--- StfCntry ,970 ,023 41,401 *** par_4 ### Significant###


Estimate We can compute R2

trstlgl <--- PolTrust ,771 ,7712= 0,594

trstplt <--- PolTrust ,810 ,8102= 0,656

trstprl <--- PolTrust ,877 ### Large values ### ,8772= 0,769

stfdem <--- StfCntry ,795 ,7952= 0,632

stfgov <--- StfCntry ,903 ,9032= 0,815

stfeco <--- StfCntry ,795 ,7952= 0,632

121

Squared Multiple Correlations: (Group number 1 - Default model). This is R2

Estimate

stfeco

,631 = ,7952

stfgov

,816 = ,9032

stfdem

,632 = ,7952

trstprl

,769 = ,8772

trstplt

,656 = ,8102

trstlgl

,595 = ,7712



PolTrust <--> StfCntry 3,281 ,129 25,485 *** par_5 ### Significant ###


Estimate

122

Estimate

PolTrust <--> StfCntry ,843 ### Smaller than 1 ###



PolTrust

4,627 ,181 25,579 *** par_6

StfCntry

3,275 ,147 22,256 *** par_7

e6

2,513 ,090 27,853 *** par_8

e5

1,750 ,068 25,858 *** par_9 ###Positive ###

e4

1,393 ,070 19,843 *** par_10

e9

1,907 ,070 27,400 *** par_11

e8

1,020 ,059 17,269 *** par_12

e7

1,800 ,066 27,421 *** par_13


StfCntry PolTrust stfeco stfgov stfdem trstprl trstplt trstlgl

StfCntry 1,000

PolTrust ,843 1,000

123


stfeco ,795 ,670 1,000

stfgov ,903 ,761 ,718 1,000

stfdem ,795 ,670 ,632 ,718 1,000

trstprl ,739 ,877 ,587 ,667 ,587 1,000

trstplt ,682 ,810 ,542 ,616 ,542 ,710 1,000

trstlgl ,650 ,771 ,516 ,587 ,517 ,676 ,624 1,000

124

FIT DIAGNOSTICS

Model Fit Summary

CMIN

Model NPAR CMIN DF P CMIN/DF

Default model 13 121,665 8 ,000 15,208

Saturated model 21 ,000 0

Independence model 6 8726,410 15 ,000 581,761

RMR, GFI


Default model ,113 ,982 ,953 ,374




Model NFI

Delta1

RFI

rho1

IFI

Delta2

TLI

rho2 CFI

Default model ,986 ,974 ,987 ,976 ,987


1,000

1,000


125

RMSEA


Default model ,078 ,066 ,091 ,000


AIC

Model AIC BCC BIC CAIC

Default model 147,665 147,743 222,462 235,462

Saturated model 42,000 42,127 162,826 183,826

Independence model 8738,410 8738,446 8772,932 8778,932

126

Matrices (Group number 1 - Default model

Residual Covariances (Group number 1 - Default model)


stfeco ,000

stfgov ,003 ,000

stfdem ,106 -,053 ,000

trstprl -,177 ,010 -,128 ,000

trstplt -,136 ,138 -,062 ,052 ,000

trstlgl ,055 ,007 ,374 ,021 -,136 ,000



stfeco ,000

stfgov ,023 ,000

stfdem ,860 -,387 ,000

### Values < 2 ###

trstprl -1,356 ,072 -,956 ,000

trstplt -1,161 1,066 -,515 ,367 ,000

trstlgl ,426 ,047 2,827 ,140 -,990 ,000

Modification Indices (Group number 1 - Default model)

127


M.I. Par Change

e7 <--> StfCntry 8,471 ,111

e7 <--> PolTrust 12,506 -,164

e8 <--> PolTrust 5,674 ,096

e9 <--> e7 10,516 ,146

e9 <--> e8 6,049 -,096

e4 <--> StfCntry 8,131 -,106

e4 <--> PolTrust 5,407 ,099

e4 <--> e7 5,200 -,101

e4 <--> e9 12,892 -,164

e5 <--> e7 7,757 -,125

e5 <--> e8 22,745 ,193

e5 <--> e9 6,924 -,122

e5 <--> e4 4,317 ,088

e6 <--> StfCntry 8,911 ,135

e6 <--> PolTrust 6,250 -,134

e6 <--> e8 19,909 -,211

e6 <--> e9 62,688 ,427

e6 <--> e5 13,862 -,193


If you repeat the analysis treating the covariance

between e7 and StfCntry as a free parameter, its

estimate will become larger by approximately 0,111

than it is in the present analysis.

PROBLEM:

Change in covariances, the change in correlations is

not known!

128

M.I. Par Change

stfeco <--- trstprl 4,641 -,027

stfeco <--- trstplt 6,938 -,036

stfgov <--- trstplt 11,171 ,041

stfdem <--- trstlgl 22,102 ,059

trstprl <--- stfeco 5,041 -,031

trstprl <--- stfdem 8,843 -,040

trstplt <--- trstlgl 4,962 -,027

trstlgl <--- stfdem 28,944 ,085

129

Model modification:

130




trstlgl

trstplt

trstprl

stfdem

stfgov

stfeco


PolTrust

e6

e5

e4

StfCntry

e9

e8

e7







131




stfeco 4,885

stfgov 3,734 5,534

stfdem 3,284 3,792 5,182

trstprl 3,007 3,861 3,152 6,020

trstplt 2,565 3,405 2,721 3,977 5,080

trstlgl 2,898 3,446 3,304 4,153 3,370 6,204



stfeco 1,000

stfgov ,718 1,000

stfdem ,653 ,708 1,000

trstprl ,554 ,669 ,564 1,000

trstplt ,515 ,642 ,530 ,719 1,000

trstlgl ,526 ,588 ,583 ,680 ,600 1,000

132








Chi-square = 56,072



133





trstlgl <--- PolTrust ,882 ,020 43,145 *** par_1

trstplt <--- PolTrust ,846 ,018 46,790 *** par_2



stfgov <--- StfCntry 1,196 ,026 46,064 *** par_3

stfeco <--- StfCntry ,979 ,024 41,131 *** par_4


Estimate


trstplt <--- PolTrust ,812


stfdem <--- StfCntry ,787

stfgov <--- StfCntry ,910

stfeco <--- StfCntry ,793

134



PolTrust <--> StfCntry 3,229 ,128 25,315 *** par_5

e6 <--> e9 ,448 ,057 7,815 *** par_6


Estimate

PolTrust <--> StfCntry ,835

e6 <--> e9 ,199



PolTrust

4,674 ,182 25,727 *** par_7

StfCntry

3,202 ,146 21,860 *** par_8

e6

2,581 ,092 28,080 *** par_9

e5

1,732 ,068 25,635 *** par_10

e4

1,346 ,071 19,050 *** par_11

e9

1,969 ,072 27,270 *** par_12

e8

,955 ,060 15,814 *** par_13

e7

1,816 ,067 27,214 *** par_14

135


Estimate

stfeco

,628

stfgov

,827

stfdem

,619

trstprl

,776

trstplt

,659

trstlgl

,585

Matrices (Group number 1 - Default model)

Implied (for all variables) Covariances (Group number 1 - Default model)


StfCntry 3,202

PolTrust 3,229 4,674

stfeco 3,134 3,161 4,885

stfgov 3,828 3,862 3,748 5,534

stfdem 3,202 3,229 3,134 3,828 5,171

trstprl 3,229 4,674 3,161 3,862 3,229 6,020

trstplt 2,733 3,956 2,676 3,268 2,733 3,956 5,080

trstlgl 2,847 4,121 2,787 3,405 3,295 4,121 3,488 6,214

136



StfCntry 1,000

PolTrust ,835 1,000

stfeco ,793 ,662 1,000

stfgov ,910 ,759 ,721 1,000

stfdem ,787 ,657 ,624 ,716 1,000

trstprl ,736 ,881 ,583 ,669 ,579 1,000

trstplt ,678 ,812 ,537 ,616 ,533 ,715 1,000

trstlgl ,638 ,765 ,506 ,581 ,581 ,674 ,621 1,000

Implied Covariances (Group number 1 - Default model)


stfeco 4,885

stfgov 3,748 5,534

stfdem 3,134 3,828 5,171

trstprl 3,161 3,862 3,229 6,020

trstplt 2,676 3,268 2,733 3,956 5,080

trstlgl 2,787 3,405 3,295 4,121 3,488 6,214

137

Implied Correlations (Group number 1 - Default model)


stfeco 1,000

stfgov ,721 1,000

stfdem ,624 ,716 1,000

trstprl ,583 ,669 ,579 1,000

trstplt ,537 ,616 ,533 ,715 1,000

trstlgl ,506 ,581 ,581 ,674 ,621 1,000



stfeco ,000

stfgov -,105 ,000

stfdem 1,223 -,268 ,072

trstprl -1,188 -,001 -,576 ,000

trstplt -,949 1,056 -,102 ,148 ,000

trstlgl ,865 ,293 ,060 ,215 -,861 -,058

138

Modification Indices (Group number 1 - Default model)


M.I. Par Change

e7 <--> StfCntry 5,054 ,085

e7 <--> PolTrust 7,595 -,129

e8 <--> StfCntry 4,415 -,066

e8 <--> PolTrust 7,139 ,107

e9 <--> e7 13,443 ,164

e4 <--> StfCntry 5,354 -,085

e4 <--> e7 5,321 -,102

e5 <--> e7 8,455 -,131

e5 <--> e8 13,574 ,148

e6 <--> e7 7,879 ,146

e6 <--> e5 8,934 -,153


M.I. Par Change

139


M.I. Par Change

stfeco <--- stfdem 6,196 ,033

stfeco <--- trstplt 6,111 -,033

stfgov <--- trstplt 8,429 ,035

stfdem <--- stfeco 4,332 ,029

trstprl <--- stfeco 4,489 -,029

trstlgl <--- stfeco 4,035 ,032

Model Fit Summary

CMIN

Model NPAR CMIN DF P CMIN/DF

Default model 14 56,072 7 ,000 8,010

Saturated model 21 ,000 0

Independence model 6 8726,410 15 ,000 581,761

140

RMR, GFI


Default model ,074 ,992 ,976 ,331




Model NFI

Delta1

RFI

rho1

IFI

Delta2

TLI

rho2 CFI

Default model ,994 ,986 ,994 ,988 ,994


1,000

1,000


RMSEA


Default model ,055 ,042 ,069 ,252


141

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Nunnally, J.C. and Bernstein, I.H. (1994), Psychometric Theory, 3rd ed., McGraw-Hill, New York, NY. Ribbink, D., van Riel, A., Liljander, V. and Streukens, S. (2004), “Comfort your online customer: quality, trust and loyalty on the internet”, Managing Service Quality, Vol. 14, No. 6, pp. 446-456 White, J.C., Varadarajan, P.R. and Dacin, P.A. (2003), “Market situation interpretation and response: the role of cognitive style, organizational culture, and information use”, Journal of Marketing, Vol. 67, No. 3, pp. 63-79. Wolfinbarger, M., Gilly, M.C., 2003. ETailQ: dimensionalizing, measuring and predicting retail quality. Journal of Retailing 79 (3), 183–198.

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Introduction to Structural Equation Modeling with Amos … · Structural Equation Modeling with Amos ... Confirmatory Factor Analysis ... Factor Analysis: analisis of covariances

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