1

1 2

Introduction to Structural Equation Modelswith Latent Variables

Josep AllepúsBenevento, May 5 th 2004

3

Scope

v Introduction to Structural Equation Models with Latent Variables

v Measuring errors in Social Sciences. The properties of the

measuring instruments used (reliability and validity)

v Correlation and Causality

v Models for the study of causality. Link between causal relations

and covariances: Path analysis. Covariance structure models

v Multivariate Analysis

v Exploratory Factorial Analysis

v Confirmatory Factor Analysis

v Measurement of quality in services. Perceived Quality’

Attributes. Cuestionario SERVQUAL

v Measuring a construct. Questionnaires

4

Statistics in Social Sciences

The two fundamental types of problems encountered in

social and behavioural sciences are:

vThe core of the problem to be studied:

The relationship between the variables andtheir relative explanatory power(correlation and causality)

vThe propert ies of the measuring instruments used

(reliabil ity and validity)

5

Measuring errors in Social Sciences

Measuring errors are omnipresent and have a tremendous influence on

statistical relations increasing interested arise on how data are collected

and measured.

Ø“To err is human,

to forgive, divine –

but to include errors in your desing is statistical”

Leslie Kish “Chance, Statistics and Statisticians”

ØThe need to include error in our statistical analysis

6

Types of Measurement Error

v Systematic Measurement Errors

ØConstant Bias

ØVariable Bias

v Stochastic Measurement Errors

ØErrors in the Dependent Variable

ØErrors in the I n d e p e n d e n t V a r i a b l e s

2

7

Objectives

v Questionnaires as instrument of measure

v Introduce basic concepts and instruments for deal ing

with measurement errors

v Show part icipants how to take measurement errors

into account when specifying and analysing empir ical

data

v Int roduce a methodology of Structural Equation

Models :

Ø that relate variables measure d with errors

Ø for handl ing latent var iab les

v Int roduce LISREL program

8

Linear relationships

In order to quantify the degree of l inear relationship

between var iables, the sample covar iance can be

used :

v A correlat ion is a covar iance between two

standardized variables (with unit variance).

v Is computed as the covariance divided by the product

of standard deviations.

v Covariances and correlat ions are not appropriate

when nonlinearit ies or outl iers are present.

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Correlational Study

v Describes an existing relationship between variables

A major purpose of correlational research is to clarify ourunderstanding of important phenomena by identifying

relationships among variables

Ê explanatory studies

v Quantifies the degree to which two or more quantitative

variables are related

Ê Uses correlation coefficient

v range from 0 to 1

Østrength (magnitude of the relationship) Lies between –1 y 1.

Positive- high scores from one variable

à high scores from the other variableØNegative- high scores on one variable

à low scores on the other variablev Can make predictions with varying degrees of confidence

ØThe stronger the correlation the greater predictionr value

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Correlation and Causality

Conclusions Drawn from Correlat ions:

CANNOT infer causal i ty from correlat ions!!!

Why?

v Don’t know which variable is influencing the other

v Could be a third, extraneous variable which would

be interpreted as a spurious relat ionship

v Time-order relat ionships (direct ion of cause)

v Spurious correlat ions

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Partial Correlation

Race IQ

SESIndicates the r betweenRace and IQ

The relationship doesn’t lookas strong now that SES is partialled out.

12

Path Analysis

Despite what your book says, you CANNOT infer causality from

these models…

Mediator variable: explains the correlation between two variables.

Moderator: affects direction or strength of correlation between two

variables.

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13

Models for the study of causality

v   Analysis of variance (1920-1930): decomposition of the variance of a dependent

variable in order to identify the part contributed by an explanatory variable(dependence analysis). Control of third variables.

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

v   Path analysis (1920-70): analysis of correlations (interdependence). Otherwise

similar to econometric models.

v   Factor analysis (1900-1970): analysis of correlations among multiple indicators of

the same variable. Measurement quality evaluation.

v   SEM (1970): Goldberger organizes a multidisciplinary conference where

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. SEM are well suited for microeconometrics

(individual data).

Aggregated data have smaller measurement errors but other types of problems

(autocorrelation) solved by dynamic macroeconometrics (1970-90).

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SEM: 5 Cs

Optimal estimation methods, robust testing procedures and goodness

of fit indices, modelling strategies, SEM are nowadays very popular

(in some journals around half of all published articles use them)

because they make it possible to (5 Cs):

1) Work with Constructs measured through indicators, and evaluate

measurement quality.

2) Consider the true Complexity of phenomena, thus abandoning uni

and bivariate statistics.

3) Conjointly consider measurement and prediction, factor and path

analysis, and thus obtain estimates of relationships among

variables that are free of measurement error bias.

4) Introduce a Confirmatory perspective in statistical modelling. Prior

to estimation, the researcher must specify a model according to

theory.

5) Decompose observed Covariances, and not only variances, from

an interdependence analysis perspective.

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Measurement of quality in services

v Services have immaterial components, which make it necessary

to take the customer’s view into account in order to evaluate

quality.

v Parasuraman et al. (1985, 1988, 1991) define quality as the gap

between consumers’ expectations prior to the service delivery

and consumer perceptions during the service delivery.

v Parasuraman et al. defined 5 aspects of any service, which can

cause a discrepancy between expectations and perceptions and

they elaborated the SERVQUAL questionnaire.

v Other authors show that relevant aspects can differ from service

to service.

v It has been suggested that perceptions already imply a

comparison with some sort of ideal.

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Perceived Quality’ Attributes

E.Tangibles

Fiabilidad

C.Respuesta

Empatía

Seguridad

Parasuraman, Zeithaml, & Berry

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CONSTRUCTOδ1

Measuring a construct

One unobservable factor andone indicator

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∑

σγσ

γσζ+σγ=θ

xxxx

xxxx2 )(Var

)(

)x,y(Covxxxy

yxyy =

σσσσ

=∑

Example: Simple regression

What are SEM? The regression model from a different perspective.

Simple regression: y = γx + ζ

= Implied Covariance Matrix

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Syllabus SEM

v Jöreskog and Sörbom developed LISREL

ØMatrices: λx θδ λy θε Ψ Φ β ΓØVariables: X Y η ξ ζØIntercepts: τ κ

v Development of confirmatory factor analysis

(Jöreskog, 1969) led to :

• objective, statistically based decisions as toadequacy

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Path Diagrams

Variable

Causal Arrow

Correlational Arrow

x y

Correlation

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Path Notation

v Relationships

Østraight arrow (causal)

Øcurved arrow

(unspecified)

v Variables

Øcircles vs. squares

ØX exogenous

(independent)

ØY endogenous

(dependent)

v Errors

Øone for everyendogenous variable

Øunexplained componentof predicted variables

Xη

Y1

Y2

δ1

δ2

H

22

x

y

Regression

x

x

Doing “Normal” Statistics

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Latent Variables

v The true power of SEM comes from latent variable modelling

v Measuring abstract concepts is central to testing Social Sience

theories

Ê Theories and concepts are of little use if they cannot be

measured and tested

Variables in Social Sciences are rarely measured directly

Øthe effects of the variable are measured

• Intelligence, self-esteem, depression• Reaction time, diagnostic skill

v Latent variables are drawn as ellipses

Øhypothesised causal relationship with measured variables

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Measuring a Latent Variable

x = t + e

v Reliability is:

• the square root of proportion of variance in x that isaccounted

• the correlation between x and e

MeasuredTrue Score

Error

5

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The univariate consequences ofmeasurement error

v Measured var iable has two causes

Ølatent variable

Ø“other stuff” random error

v x = True Score + Error = ξ + δ

v ⇒ Var(x) = V a r(ξ) + Var(δ) = φ + θ

v Thus, Var(x) overestimates the variance of the true

score

CONSTRUCTOδ1

27

Multi-Indicator Approach

vA mult iple-indicator approach reduces the overal l

ef fect of measurement error of any individual observed

variable on the accuracy of the results

vA dist inction is made between observed variables

(indicators) and underlying latent variables (constructs)

vTogether the observed variables

and the latent variables make

up the measurement model

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Preparing the Instrument

v The most common types of instruments used are the

questionnaire (self-administered)

v Either type of instrument should consist of questions

that can be asked exactly the way i t is writ ten,

quest ions that wi l l mean the same thing to everyone

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Analysing Scale Data

vWhen construct ing a psychometr ic scale you need to

Øcheck the faci l i ty of i tems

Øcheck the reliabil i ty of the scale

Øcheck the construct validity of the scale

v To examine the sub-scale structure

v To confirm the blue-print structure built into the scale

v A method for s impl i fy ing and summariz ing the

correlation matrix

v Constructs “components” or “ factors” which capture

the systematic inter-correlation of the items

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Multivariate Analysis

1) Adecuación de los datos para la

aplicación del modelo factorial

2) Análisis factorial exploratorio

2.1 Extracción de los factores

2.2 Rotación de los factores encontrados

2.3 Interpretación

3) Análisis factorial confirmatorio

4) Validación del modelo

Adecuación de los datos para el análisis

Matrix correlations R

v El determinante de dicha matriz |R| t iene un valor muy próximo

a cero.

=1rrrr

...

...

1rrr

1rr

1r

1

R

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Exploratory Factorial Analysis

F1

F2

Two unobservable factor andfour indicators

6

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Link between causal relations and covariances:Path analysis

Types of relat ionships among variables:

vPath analysis decomposes covar iances in order to seek

information about underlying causal relat ionships.

vWith this goal in mind, one must start in the opposite

direction: deriving covariances from the parameters of the

causal process.

vDrawing a “path d iagram” is the first stage in path

analysis.

33

Covariance structure models

SEM are also called covariance structure models.

The structural equation system expresses each element of the population

covariance matrix Σ as a function of model parameters. These model

parameters thus impose a structure on Σ.

Σ= Σ (π) where:

ØΣ: Population covariance matrix (with variances on the main

diagonal).

Øπ: vector containing all parameters (e.g. effects, disturbance

variances, variances and covariances of exogenous variables).

Path analysis is useful for obtaining an insight into a causal process and

into the possible effects explaining a covariance. Unfortunately this

information is often insufficient. Many models can explain the same set of

covariances. The choice among them cannot be statistical but theoretical.

34

Correlation and causality

The falsification principle (Popper, 1969) corresponds to what logic

calls “modus tollens”.

vA hypothesis is rejected if its consequences are not observed in

reality. Thus, causal theories can be rejected (falsified) if they are

contradicted by data, that is, by covariances and correlations.

vOn the contrary, theories cannot be statistically confirmed. A

correlation can be due to a causal relation or to many other

sources.

When studying the relationship between two variables, non-

experimental research cannot control (keep fixed) other sources of

variation. For this reason, all relevant variables must be in the

model.

35

Data preparation

v Enter raw data

v Run Frequencies into SPSS to check for any out-of-range or

missing data

v Identify items that have been “reversed” and recode them so

that all expected item-item correlations will be positive

v (Use Recode into Same variables, and do them all together as a

batch)

Descriptives:

v The first step is to examine the means and standard deviations

of each item

v Omit items with a mean that is too high or too low (facility)

v Omit items with low standard deviations (not discriminating)

36

Correlations

v Generate the correlat ion matr ix

v Check for any correlat ion values that are very high

( indicates two i tems that have the same meaning)

37

PRELIS

It has the following functions:

v Exploratory data analysis: histograms and normali ty

tests.

v Computat ion of S and Γ.

v S may be a covariance matr ix or a correlat ion matr ix.

v Computat ion of an appropriate Γ matrix for WLS

estimation or, even better, for correct standard errors

and test statistics.

7

38

Adecuación de los datos para el análisis

v Matrix correlations R

v El determinante de dicha matriz |R| t iene un valor

muy próximo a c e r o .

v El Test de esfericidad de Bartlett

v La matriz de correlación anti- imagen

v Las medidas de adecuac ión de la muestra

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Adecuación de los datos para el análisis

v Matrix correlationsR

v El determinante de dicha matriz |R| t iene un valor

muy próximo a c e r o .

v El Test de esfericidad de Bartlett

v La matriz de correlación anti- imagen

v Las medidas de adecuac ión de la muestra

v El índice de Kaiser-Meyer-Olkin

40

Adecuación de los datos para el análisis

v Matrix correlationsR

v El determinante de dicha matriz |R| t iene un valor

muy próximo a c e r o .

=1rrrr

...

...

1rrr1rr

1r

1

R

41

Adecuación de los datos para el análisis

v Matrix correlationsR

v El determinante de dicha matriz |R| t iene un valor

muy próximo a c e r o .

v El Test de esfericidad de Bartlett

v La matriz de correlación anti- imagen

=

...

...

1000100

101

R:0H

42

Adecuación de los datos para el análisis

v Matrix correlationsR

v El determinante de dicha matriz |R| t iene un valor

muy próximo a c e r o .

v El Test de esfericidad de Bartlett

v La matriz de correlación anti- imagen

v Las medidas de adecuac ión de la muestra

∑∑∑

≠≠

≠

+=

ji

2ij

ji

2ij

ji

2ij

jar

r

MSA

43

Raw Data

=Σ

=

]'X·X[E1rrr

1rr1r

1

R

X1 X2 X3 X4 1) - - - -2) - - - -3) - - - -

.....865) - - - -Y

Covariance Matrix to be Analyzed EEobs X1 X2 X3 X4 ------- ------- ------- -------X1 63.3820X2 70.9840 110.2370X3 41.7100 52.7470 60.5840X4 30.2180 37.4890 36.3920 32.2950

8

44

Exploratory Factor Analysis

F1

F2

45

EFA Linear Equations

F1

F2

X1 = 8811 @@F1 + 8812 @@F2 +...+ 881m @@Fm + **1

....

X = 77·F+**

46

Exploratory Factor Analysis

What Confirmatory Factor Analysis can do

that Exploratory Can’t do:

v Introduce genuine restr ict ions rather than statist ical

and arbitraty o n e s

v Structure the data in accordance with specif ic model

paramenters for a concrete appl icat ion

v Achieve a single soluct ion for these parameters

v Check the val idity of the theory that enabled you to

construct the model

47

Confirmatory Factor Analysis Model

δ+Λ= FX ·Where:

X = (q × 1) vector of indicator/manifest variables

F = (n × 1) vector of latent constructs (factors ξ)

δ = (q × 1) vector of errors of measurement

Λ= (q × n) matrix of factor loadings

49

Matrix Symbolism

X16.75

X221.26

X310.03

X46.10

F1 1.00

F2 1.00

7.53

9.43

7.11

5.12

0.78

X = 77·F+**

+

=

2

11F

2

1

2X

1X

δδ

λλ

+

=

4

32F

4

3

4X

3X

δδ

λλ

)(N

400000

30000

2000

100

00001

00001

,

0

0

0

0

0

0

4

3

2

1

2F

1F

=

θ

θ

θθ

φφ

δ

δ

δδ

50

Statistical constraints

X16.75

X221.26

X310.03

X46.10

F1 1.00

F2 1.00

7.53

9.43

7.11

5.12

0.78

X = 77·F+**

+

=

2

11F

2

1

2X

1X

δ

δ

λ

λ

+

=

4

32F

4

3

4X

3X

δ

δ

λ

λ

)(N

400000

30000

2000

100

00001

00001

,

0

0

0

0

0

0

4

3

2

1

2F

1F

=

θθ

θ

θφ

φ

δδδ

δ

9

51

Confirmatory Factor Analysisδ1

δ2

δ3

δ4

F1

F2

φ

Covariance Matrix Σ issymmetric and

Unknown parameters are:+4 lambda+4 delta+1 φ correlación entre los factores

102

)14(4=

+

Number of Degrees of freedom are:g = 10 - 9 = 1 g.l.

We have 10 different observed elements form Sand 9 unknown elements, thus the model is over identifiedNecessary condition for model’s identification: g $ 0

52

IdentificationvThe extent to wich we are able to estimate everything we

want to estimateÊ Can model parameters be der ived from var iances andcovariances?

vIdentif ication must be studied prior to data collection

vIn pr inciple, checks should be made to see i f for each free

parameter there exists at least one algebraic expressionlinking it to variances and covariances. A l ist of necessaryand suff ic ient condit ions may help, though not al l models can

be studied in this way.

Many models wil l not fulf i l the sufficient condit ions and wil lnevertheless be identif ied. In this case, an empirical check

can be carried out:

vThe model is est imated using these data. Al l software

programs do an ident i f icat ion test.

53

Identification conditions

1) Necessary conditions: According to d. Of freedom g, models

can be classified into:

vNever identified (g<0): infinite number of solutions for some

parameters that makes S equal Σ(p).

vPossibly identified (g=0): there may be a unique solution for all

parameters that makes S equal Σ(p).

This type of models is less interesting in that their rejection is not

possible (their restrictions are not testable).

vPossibly overidentified (g>0): there is no solution for p that

makes S equal Σ (p) but there may be a unique solution that

minimizes discrepancies between both matrices. Only these

models, more precisely their restrictions, can be tested from the

data.

54

Identification conditions: example

Number of Degrees of f reedom are:

g = 10 - 9 = 1 g . l .

We have 10 dif ferent observed elements form S

and 9 unknown elements, thus the model is over

identified

Necessary condit ion for model’s identi f ication: g $ 0

55

Existence of d.of.freedom

v A more pars imonious model ] will facilitate

interpretation

v There is not one unique parameter vector.

An “opt imum” p according with some cr i ter ion

Not al l est imat ion methods lead to the same

solut ion

v The existence of degrees of freedom enables us to

compare the hypotheses included in the model.

Ø leads to a residual matrix (S-Σ (p))

ØGoodness of f i t become necessary

56

Estimation

v First est imate the sample variances and covariances (S) and

then f ind the best f i t t ing p parameter values.

v Fit function: related to the size of the residuals in S-Σ(p), which

we now arrange as a vector by also dropping dupl icated

elements. The generic expression of the f i t function is:

F= (S- Σ(p))' W (S- Σ(p))

Ø(S -Σ(p)) ’ is a row vector of residuals,

Ø(S -Σ (p)) a column vector and

ØW the weight matrix.

10

57

Estimation Methods

Common estimation methods differ only by the choice of W:

v Unweighted least squares (ULS) with W=I.

v Normal theory weighted least squares (NT-WLS, called Generalized Least

Squares –GLS– by LISREL), with W=G-1 and G computed from S.

Asymptotically efficient under multivariate normality.

v Maximum likelihood (ML), with W=G-1 and G computed from Σ(p).

Asymptotically efficient under multivariate normality.

v Asymptotic distribution free method (ADF, called Weighted Least Squares

–WLS– by LISREL), with W=G-1 and G computed from 4th order moments.

Asymptotically efficient under any distribution. In practice it has been found

to be appropriate only for samples larger than 1000 and models with 10 or

fewer observed variables.

58

Results of the LISREL run

Path diagram window. The user can:

v Modify the layout by dragging variables to the desired posit ion.

v Select which parts of the model are shown (view, opt ions,

visible).

v Select which stat ist ical information is displayed, one at a t ime(view, estimations):

ØEstimates.

ØStandardized solution.

ØConceptual diagram: with arrows only, no values.

Øt-values or Wald tests for included parameters.

ØModification indices or Lagrange multiplier tests of omitted parameters.

ØExpected changes: approximate estimate that would be obtained if theparameter was set free.

v A text output f i le contains more information (select i t from the

window menu) .

59

Path diagram

Chi-Square=1.25, df=1, P-value=0.26272, RMSEA=0.017

21.26

10.03

6.10

F1 1.00

F2 1.00

9.43

7.11

5.12

0.78

6.757.53

60

LISREL Results: 7

X16.75

X221.26

X310.03

X46.10

F1 1.00

F2 1.00

7.53

9.43

7.11

5.12

0.78

LISREL Estimates (Maximum Likelihood)

X1 = 7.5251*F1, Errorvar.= 6.7548 , R² = 0.8934 35.1567 5.9484

X2 = 9.4330*F1, Errorvar.= 21.2562, R² = 0.8072 32.4872 10.6694

X3 = 7.1101*F2, Errorvar.= 10.0307, R² = 0.8344 33.0179 8.6446

X4 = 5.1184*F2, Errorvar.= 6.0974 , R² = 0.8112 32.3253 9.8257

61

LISREL Results: ECovariance Matrix to be Analyzed X1 X2 X3 X4 -------- -------- -------- --------X1 63.3820X2 70.9840 110.2370X3 41.7100 52.7470 60.5840X4 30.2180 37.4890 36.3920 32.2950

Fitted Covariance Matrix X1 X2 X3 X4 -------- -------- -------- --------X1 63.3820X2 70.9840 110.2370X3 41.8328 52.4388 60.5840X4 30.1143 37.7493 36.3920 32.2950

62

LISREL Results: E

La matr iz de covarianzas estimada a partir del

modelo v iene determinada por

E = E[X·X ’]

suponiendo incorrelaci ón entre el factor único δ y

los factores comunes (similar a la restr icción que

supone el modelo de regresió n)

E = 7 M 7‘ + 1*

11

63

LISREL Results: E

+

=12.511.700

0043.953.7

178.0

78.01

12.50

11.70

043.9

053.7

+

10.6000

03.100026.210

75.6

E = 7 M 7‘ + 1*

X16.75

X221.26

X310.03

X46.10

F1 1.00

F2 1.00

7.53

9.43

7.11

5.12

0.78

64

LISREL Results: ECovariance Matrix to be Analyzed X1 X2 X3 X4 -------- -------- -------- --------X1 63.3820X2 70.9840 110.2370X3 41.7100 52.7470 60.5840X4 30.2180 37.4890 36.3920 32.2950

Fitted ResidualsX1 X2 X3 X4 -------- -------- -------- --------X1 0.0000X2 0.0000 0.0000X3 -0.1228 0.3082 0.0000X4 0.1037 -0.2603 0.0000 0.0000

65

Residuals

v Analysis of residuals indicates the discrepancy

between the sample and the f i t ted

correlation/covariance matrices

ØCovariance Matrix to be Analyzed

ØFitted Residuals

v Interpretat ion does not proceed unt i l the goodness of

f i t has been assessed.

66

Goodness of fit assessment

Introduction

The fit diagnostics attempt to determine if the model is correct and

useful.

ØCorrect model: its restrictions are true in the population.Relationships are correctly specified without the omission of

relevant parameters or the addition of irrelevant ones.

ØIn a correct model, the differences between S and Σ (p) are

small and random.

ØCorrectness must not be strictly understood. A model must

be an approximation of reality, not an exact copy of it.

ØThus, 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 todistinguish statistically, they can be only distinguished theoretically.

67

Chi Square Test

v Represents a statistical test of the hypothesis that the residuals

are not statistically different from zero

v Probability greater than the chosen level of alpha is considered

to be an acceptable

v The minimum value of the discrepancy between S and Σ (p) –fit

function – obtained during estimation is used for the goodness of

fit assessment.

v Under the null hypothesis that the model is correct, a

transformation of this fit function called χ2 goodness of fit

statistic follows a χ2 distribution with g degrees of freedom and

can be used to do a likelihood ratio test. Rejection implies

concluding that some relevant parameters have been omitted.

68

The χχ2 goodness of fit statistic

v The interpretation depends on the power of the test (the probability of

rejecting false null hypotheses, i.e. of detecting omitted parameters):

ØLow: rejection implies some large specification errors. Acceptance

is inconclusive.

ØHigh: acceptance means no large specification errors. Rejection is

inconclusive.

Power of the test increases when:

ØSample size, R2, k or the number of indicators per factor increase.

ØCollinearity and the overall number of parameters decrease.

Usually Power is high (often). Researchers are usually willing to

accept approximately correct models with small misspecifications.

Quantifying the degree of misfit is more useful than testing the

hypothesis of exact fit.

12

69

The goodness of fit statistic: SRMR

v The standardized root mean squared residual SRMR:The simplest global descriptive fit indices evaluatethe size of standardized residuals (residual

correlations or residuals divided by the product ofstandard deviations).

v The SRMR does not take pars imony into account butdoes wel l at discr iminat ing between roughly correctand grossly incorrect models.

v Values below 0.05 are considered acceptable.

v I t can be improved by using the degrees of freedominstead of the number of var iances and covariances

( k + 1 ) k / 2 to take parsimony into account

70

Root Mean Square Error of Approximation

v The root mean squared error of approximation

( R M S E A )

v Introduced by Steiger (1980)

v based on χ2

v measure of approximate f i t

v Includes a correct ion for parsimony and sample size

71

Absolute Fit Indices

v Analogous to R2 in mult iple regression

v Represent the proport ion of variance in the sample

matr ix explained by the proposed model

v For GFI a value >0.9 is considered to be an

acceptable f i t

72

Goodness of fit: Results 1

Eobs - Emodelo = 1r e s

1) Absolute Fi t Indices

v Chi-square =1.25 (p=0.26)

v GFI =0.9993

v RMSR =0.137

v R M S E A =0.017

v E C V I =0.022

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Incremental fit indicesv Assess the improvement the proposed model represents over

a restr ict ive basel ine model

v Most commonly the basel ine model is one that specif ies no

factors and zero correlat ions among the observed variables

v They compare the χ2 stat ist ics of the researcher’s model and

a b a s e m o d e l that assumes that al l variables are uncorrelated

(independence model) .

ØThey usual ly l ie between 0 and 1 (1 shows a perfect f i t ) .

ØThe base χ2 statistic is usually very large.

These indices are often close to unity: in general only

values above 0.95 are accepted.

v A typical incremental f it index is the CFI

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Goodness of fit: normed fit index (NFI)

Bentler and Bonnet ’s (1980)

  NFI = ( χ2b - χ2

k )/ χ2b

χ2 corresponds to the researchers ’ model,

χ2b to the base model.

v N F I = 0 if χ2= χ2b and N F I = 1 i f χ2=0 .

v N F I always increases when adding free parameters.

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75

Goodness of fit: Results 2

Eo b s - Emodelo = 1res

2) Incremental f i t indices

v AGFI =0 .993

v NFI =0.9995

v CFI =0.9999

Bentler’s (1990) c o m p a r a t i v e f i t i n d e x ( C F I )

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Goodness of fit: take parsimony into account

Other indices do not l ie between 0 and 1. Their

absolute values are diff icult to interpret but they can

be used to compare models wi th the same var iables

and cases. They take parsimony into account:

v A I C (Akaike, 1987) y C A I C ( Bozdogan , 1987).

vWhere the n o n c e n t r a l i t y p a r a m e t e r ( N C P ) is defined

as χ2-g (zero if negative).

Values below 0.05 are considered acceptable

(Browne & Cudeck , 1993).

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Goodness of fit: Results taking parsimony into account

v AIC =19.2

v CAIC =71.1

v PNFI =0 .167

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Modelling Stages in SEM

79

Model modificationv Frequently models fail to pass the diagnostics. Fortunately, data can not only

be used to test models (confirmatory ) but also to drive their modification(exploratory).

v Model modifications, mainly based on detailed diagnostics and theoreticalknowledge, aim at improving either its fit or its parsimony.

v Model modification has some undesirable statistical consequences, especiallyif modifications are blindly done using only statistics, that is, without theory.

v Avoid adding theoretically uninterpretable parameters, no matter howsignificant.

v Even if model modification has been done carefully, modifications are basedon a particular sample. Have we reached a model that fits the population?

v The introduction of modifications that improve the fit to the sample but not tothe population is known as capitalization on chance.

v The only solution is to check that the model fits well beyond the particularsample used.

v Crossvalidation: estimation and goodness of fit test of the model on anindependent sample of the same population. If only one sample is available, itcan be split: the first half is used for model modification and the second forvalidation. Crossvalidation is successful if the model fits the second samplereasonably well.

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Random and systematic error

Reliability and validity:

v Reliability: Extent to which a measurement procedure “would”

yield the same result upon several independent trials underidentical conditions. Consistency of the measurement .

Ê In other words, absence of random measurement error

(but any systematic error would replicate)

 

v Validity: Extent to which a measurement procedure measures

what it is intended to measure, except for random measurementerror.

ÊValidity is concerned with random, but also has to do withsystematic error

ÊConfirmatory factor analysis models. Introduction toreliability and validity assessment

In practice we may need to trade off between validity andreliability

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81

Reliability

A rel iable instrument

provides consistent

resu l ts… how do we

measure reliabil i ty?

vTest - retest

vAlternate forms

vSplit half

vOdd - even

How do we increase reliabil i ty?

vIncrease the number of

items. Typically, the more items the

more reliable the test is

vLimit range restriction.

vProvide clear instructions

with unambiguous questions.

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Reliability

1) Usual approaches Reliability ( Sánchez & Sarabia , 1999):

v Same instrument at two points in time (critique: change of opinion;

hence non_identical conditions)

vSame instrument twice together (critique: memory; hence

dependence)

vSimilar instruments together (critique: non-identical conditions)

Measurement of reliability: Percentage of variance of trials explainedby its stable part.

Reliability may be considered for a single questionaire item or for asummated scale (sum or average of a set of items). Reliability of a

summated scale is higher: measurements errors of differents i temstend to cancel out.

2) Validez

3) Sensibilidad

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Clasical approaches to estimate Reliability(Carmines & Zeller, 1979):

v Correlations of two identical measurements (items of scales) at differentspoints in time (test-retest Reliability). &Assumptions needed:

ØStability (no change in true opinion)

ØIndependence (lack of memory)

v Correlations of two similar instruments (items of scales) taken at the samepoint in time (alternative form reliability). If only one scale is available, this

approach is still possible by splitting the items into two sub scales with the

same number of items (split-halves reliability). &Assumptions needed:

ØItems, scales of sub-scales are parallel

ØIndependence (instruments are different enough so that memory doesnot make responses dependent).

v Cronbach’s α: estimate the reliability of a scale obtained from all possibleinter-item correlations (internal consistncy reliability). &Assumptions:

ØItems are parallel

ØIndependence (itemts are different enough so that memory does not

make responses dependent).

]/1[1N

N 2xi

2∑−−

= σσα

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Scale Reliability

In SPSS, obtain a rel iabil i ty analysis,

Øfrom the Statistics dialog ask for

“Descript ives of Scale i f I tem deleted”

In output, examine item-total correlations for each item

Øremove those with low correlat ions (where alpha would

increase if they were omitted)

Øre-run the reliabil i ty analysis with the poor items

removed

ØMake a note of the f inal value of Alpha and the f inal

number of i tems

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Scale Reliabilityv In SPSS, obtain a reliability analysis

86

Validity

2) Validity: Extent to which a measurement procedure measures

what it is intended to measure, except for random measurement

error. ( Nunnally, 1987; Carmines & Zeller, 1979):

v Content Validation

v Criterion Validation: Comparision to an error-free measurementof the same concept

v Construct Validation: Analysis of correlations with measurementof the same and related concepts. Nowadays CFA are used.

Construct validity is the extent to which the survey measures what

it purports to measure.

How do we measure construct validity?

v convergent validity: high correlation between two assessments

of the same construct

v discriminant validity: low correlation with assessment of a

different construct Convergent validity

v Divergent validity

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87

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

v All items load on the factor they are supposed to measure.

v No error correlations are specified.

....and diagnose its goodness of fit.

v You can never be certain of validity, but a CFA model can help detect

signs of invalidity such as:

v It does not correctly reproduce the covariance matrix (additional

loadings or uniqueness correlations are needed, thus revealing mixeditems, additional necessary dimensions or method factors).

v Some variables have a unique variance that is too high to be attributedto solely random error (convergent invalidity).

v Some factors have correlations very close to unity (discriminant

invalidity).

v Some factors have correlations of unexpected signs or magnitudes

(nomological invalidity).

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Measurement of validity

v Measurement of val idity: Percentage of variance ofthe stable part explained by the concept of interest

Assuming the validity of v, its reliability is thepercentage k of var iance explained by f .

v Always fol low this golden rule:

ØEstimate reliabil ity after validity has beendiagnosed.

ØTest the specif icat ion of measurement equat ionsin a CFA model prior to specifying equationsrelat ing factors. Otherwise, relat ionships among

factors might be biased (specif ication errors) oreven meaningless ( inval idi ty).

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External Validity

v Another word for External Val idi ty is generalizability

90

Cuestionario SERVQUAL

Tangibles

Fiabilidad

C.Respuesta

Seguridad

Empatía

91

etc...