Transcript Structural Equation Models - Statistics for Marketing & Consumer

Structural equation
modeling
Chapter 15
Statistics for Marketing & Consumer Research
Copyright © 2008 - Mario Mazzocchi
1
Structural equation modeling
• Structural Equation Modelling (SEM) is a
powerful method to estimate multiple and
simultaneous relationships involving several
dependent variables and explanatory
variables, and allows for the inclusion of
latent variables which cannot be directly
measured but can be expressed as a function
of other measurable variables
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SEM
SEM extends the regression assumptions as
• Several dependent variables can be considered at
the same time
• Explanatory variables can be assumed to be
measured with a random error
• Endogenous variables can be used to explain
dependent variables
• Correlation between explanatory variables is
allowed for
• And this is not all. Another key feature in SEM is
the possibility of including in the model, as
endogenous or exogenous variables, some latent
(unobservable) variables.
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Latent variables
• A latent variable cannot be observed or
directly measured as it quantifies an object
or construct which cannot be defined and
measured in an unequivocal and verifiable
way
• Latent variables are not directly measured
but can be approximated by a set of
observable variables
• Factors or principal components are latent
variables
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SEM and latent variables
• SEM allows one to incorporate FA into a regression
model, where latent factors appear in the SEM as
explanatory and/or dependent variables.
• SEM can be seen as a broad model which embodies
regression and FA and canonical correlation
analysis (the relationship between sets of
dependent and independent variables are
estimated)
• However, many issues need to be considered to
estimate SEMs
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SEM as confirmatory FA
• SEM is so flexible that it may lead to
hundreds of potential alternative models
• This is the reason why SEM is classified as a
confirmatory rather than exploratory
technique (like FA) and is sometime referred
to as confirmatory factor analysis
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Exploratory FA
• Exploratory techniques
•
•
•
Get the data
Specify a parametric structure
Use an estimation procedure to quantify the
relationship among variables
• Exploratory FA
•
•
•
Assumes a linear structure and defines the factors as
unobservable variables
Factors are expressed as linear combinations of the
observed variables
Factor loadings and the factor scores are estimated,
together with some of the diagnostics (mainly the
eigenvalues) which allow ne to decide on the number
of factors
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Confirmatory FA
Confirmatory techniques
• The specific relationship among the variables need to be
specified prior to the analysis, based on some pre-existing
theory
• The technique is used to fit this pre-specified model to the
data and diagnostics are used to assess whether the model
is good or not.
SEM as confirmatory FA
• It requires the model and all of the relationships between
the variables to be chosen prior to estimation
• With confirmatory analysis it is still possible to compare
statistically alternative models through a competing model
strategy using some statistical criterion to choose the
winning mode
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Latent constructs and indicators
• Latent constructs cannot be directly and univocally
measured
• E.g. attitudes, passion, trust, risk aversion
• There is no unique measure because they have many
dimensions
• Researchers usually exploit several items in a questionnaire
to obtain some indirect measurement
• E.g. measure risk aversion by asking respondents to rate how
dangerous they consider driving at 100mph, diving from ten meters,
swimming in the ocean, jumping off a plane with a parachute, etc.
• These are indicators (or manifest variables) which help
measuring the construct, although with some measurement
error
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Causality
• Constructs are the foundation of structural equation
modeling, because they represent the real partners in a
causal relationship
• Causation: when two constructs are significantly associated
(correlated) to one another but one occurs before the other
then the construct occurring earlier is said to cause the
construct provided there are no other reasons for such
result
– E.g. a high risk aversion precedes the decision to avoid risky actions,
so risk aversion causes behaviour.
• It is debatable whether statistics are able to identify causal
relationships
• In practise causation from the explanatory variable to the
dependent one is simply hypothesized by the researcher
and statistical models merely assess whether the assumed
relationships fits the data appropriately
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From variables to constructs
•
•
•
If the true risk aversion were measurable then it would be
possible to explore its role in determining some specific
behavior like purchasing chicken when newspapers report
stories about bird flu
It would be a nonsense to relate the fear of flying to
chicken consumption
But it is no nonsense to:
•
•
•
use the fear of flying and ten other risk aversion measures to
quantify the construct of risk aversion
relate the construct to chicken consumption
SEM moves from variables to constructs because the latter
can be seen as the outcome of several measurable
variables.
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Variables in SEM
• Manifest variables that can be directly measured and serve
as indicators
• Latent variables are not observable and are the actual
components of the causal relationship
• Just like variables in regression analysis one can distinguish
between exogenous constructs and endogenous constructs
in a causal relationship
• Exogenous constructs only act as explanatory factors and
do not depend on any other construct
• Endogenous constructs play the role of the dependent
variables and they appear on the left-hand side in at least
one of the SEM causal relationships
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Examples of marketing applications for
SEM
• Scarcity and willingness to buy How perceived scarcity influences
willingness to buy through a set of mediating variables, like perceived
quality, perceived monetary sacrifice and perceived symbolic benefits (Wu
and Hsing (2006))
• E-shopping On-line shopping as the interaction of four variables: education,
technological know-how, economic condition and trust (Mahmood et al.
(2005))
• Customer relationship Customer satisfaction for credit cards considering
two latent constructs, customer service and card member rewards as
measured by some manifest variables, like being courteous, accurate in
answering questions, correcting errors (for customer service) or ease of
obtaining rewards (Dillon et al. (1997))
• Human brand . Why people become so attached to famous personalities who
are the subject of marketing communication? Three latent determinants are
considered, referring to the extent that the personality is perceived to fulfil
a consumer’s (a) autonomy (or self-determination), (b) relatedness (sense of
closeness with others) and (c) competence (the tendency to seek for
achievement (Thomson (2006))
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SEM & related techniques
• Structural Equation Modelling is a
comprehensive method which includes as
special cases:
– confirmatory factor analysis
– path analysis
– multivariate regression (simultaneous equation
systems)
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Confirmatory factor analysis
• In CFA the number of factors and the loadings of
the original variable are assumed to follow some
prior theory
• The researcher runs the factor analysis on the basis
of these assumptions on the number of factors and
the loadings (constraining to 0 the loadings for
those variables that are not expected to load on a
specific factor)
• Results are then evaluated with goodness-of-fit
diagnostic
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Path analysis
• It generalizes the regression model to deal with the
causality concept
• Path Analysis is based on the path diagram
• In Path Analysis all variables are directly measured, which
marks the distinction with SEM, which includes latent
constructs
• The path diagram represents the relationship between the
variables through arrows and boxes:
– boxes are the variables
– straight arrows leave the boxes containing predictors and point
towards the boxes containing the dependent variables
– It is also possible that two variables are correlated without implying
causation, in which case the arrows are curved.
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Example of path diagram
• A model where x1 and x2 cause y and there is nonzero correlation (there is collinearity) between x1
and x2
x1
0.7
0.3
These are correlations between
the variables linked by the
arrows
Thisy is the direct effect of x1 on y
x2
0.5
The indirect effect of x1 on y is
given by 0.3x0.5 = 0.15
The total correlation of x1 and y is 0.85
Similarly, the total correlation of x2 and y is 0.71 (0.5+0.3x0.7)
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Multivariate regression
• Multivariate regression analysis or simultaneous equation system
• a system of regression equation
• the dependent variable in one equation can appear on the right hand side
of other equations
• Some of the key assumption of standard regression analysis are violated:
• the endogenous variables which appear on the right hand side are correlated
with the residuals
• Least squares estimates are inconsistent – alternative estimation methods are
used
• SEM as a system of structural equations
• The equations of a system can be defined as structural equations when
their specification depends on some theoretical basis
• The structural form can generally be simplified into a reduced form by
solving the system analytically
• Differences
• SEM allows for the inclusion of latent constructs (unobserved factors) and
uses a confirmatory approach
• Multivariate regression is based on observed (manifest) variables only and is
generally exploited for exploratory rather than confirmatory analysis
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Structural equation models
• A structural equation model is composed of
• a measurement model which links the latent construct to the
manifest indicators; and
• a structural model which summarizes the relationships linking the
endogenous and exogenous constructs
• The measurement model corresponds to confirmatory factor analysis:
it can be tested whether the measurement of the latent variables
through the manifest indicators is satisfactory
• A structural equation model can be represented through a
path diagram
• manifest variables are shown in square or rectangular boxes
• latent variables (and measurement errors) are shown
through ovals or circles
• causality relationships are indicated through straight arrows
• correlation without causality is shown through a curved
arrow
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Identification
• The path diagram is drawn according to some theory
• Quantities to be estimated
• Latent constructs
• Parameters (causal relationships and correlations)
• Before estimation it is important to check for identification
issues
• A model can be
• Under-identified when the unknown parameters are too
many compared to the available observations
• just-identified when the model allows a single solution since
there is a single set of estimates compatible with the
available observations
• over-identified when a single structural model is compatible
with various sets of estimates (thus estimates might be not
unique nor optimal)
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Over-identification and SEM
• Signs of identification problems
• standard errors look too large
• indicators are too highly correlated between each other
• some of the estimates are unacceptable (like negative
variance)
• If the number of relationships exceeds what is needed for
just-identification there is over-identification
• SEM allows one to test over-identified models and find a
solution which is statistically acceptable and test the
validity of a theory (or compare competing theories)
• To check for identification and over-identification, one may
look at the degrees of freedom
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Degrees of freedom (DoF)
• DoF count the number of independent observations
available for estimation
• They are a measure of the discrepancy between
the available number of observations and the
constraints associated with the estimation of
unknown pardegrees of freedomameters
• Under-identified models: negative degrees of freedom
• Just-identified models: degrees of freedom equal to zero
• Over-identified models: positive
• Rule of thumb for identification. At least three
manifest indicators for each latent variable.
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Estimation
• Maximum likelihood estimation (MLE)
• the manifest indicators must follow a multivariate normal distribution (i.e.
they are normally distributed for any value of the other indicators)
• the latent constructs are also assumed to be normally distributed
• Key aspects of SEM estimation
• Individual cases only enter the estimation process to obtain the covariance
matrix
• SEM does not use the individual observations (cases) for the estimation of
the parameters
• Estimation is based on the covariance matrix not on the individual cases
• Thus, degrees of freedom refer to the elements in the covariance
matrix
• An adequate sample size is still needed
• Identification problems may emerge when many of the elements of the
covariance matrix are close to zero unless the sample size is large enough
• A simple rule of thumb requires at least fifteen observations per measured
variable or indicator
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Testing
• When there are more elements in the
covariance matrix than parameters to be
estimated, then the model is over-identified
and it is possible to test its theoretical
foundation
• How is the theory tested?
• First, parameter estimates should be
reasonable, both in terms of the founding theory
and for statistical acceptability
• Goodness-of-fit tests
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Goodness-of-fit tests
• Chi-square statistic: tests whether the observed covariance
matrix is equal to the estimated one (which is what one
hopes)
• The number of degrees of freedom indicates whether the model is
just-identified (zero DoF) or over-identified (more than zero DoF)
• If the p-value of the Chi-square test is larger than 0.05 (0.01), then
the observed covariance matrix is not different from the estimated
one at a 95% (99%) level of confidence
• Non-rejection of the null hypothesis suggests that the theory is
acceptable although this does not rule out better models
• The output from the tested model is usually compared to
two boundaries
• the independence model (no correlation between the endogenous and
exogenous variables)
• the saturated model (no constraints at all, perfect fit with the data,
just like in log-linear analysis).
• The tested model lies between these two extremes
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Goodness-of-fit indicators
• Minimum sample discrepancy (CMIN) checks whether the model
perfectly fits the data (very unlikely and not really useful as a test).
• When this measure is divided by the degree of freedom (CMIN/DF), one
obtains the above mentioned Chi-square test
• Root mean square residual (RMR) refer to the residuals between the
estimated and sample covariance matrices. It can be used to compare
alternative models, where a smaller RMR indicates better fit
• Goodness-of-fit index (GFI), should be above 0.90 for acceptable
theories. The adjusted version (AGFI) has similar interpretation
• Other indices which are expected to be as close as possible to one (and
not below 0.90):
•
•
•
•
•
normed fit index (NFI)
relative fit index (RFI)
incremental fit index (IFI)
Tucker-Lewis coefficient (TLI)
comparative fit index (CFI)
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More indicators and tests
• The non-centrality parameter (NCP) and the root mean
square of approximation (RMSEA) consider both the
discrepancy criterion (like the CMIN) and some parsimony
criteria accounting for degrees of freedom
– The RMSEA should be less than 0.05 for a good model
– The hypothesis that RMSEA<0.05 is tested through the PCLOSE test.
• Other measures for comparing alternative models are the
AIC and BIC information criteria and similar information
indices
• The Hoelter’s critical N shows the largest sample size
which still allows to accept the model
– It is a useful complement to the Chi-square test, as the latter tends
to reject the model when the sample size is large.
– Better models require larger sample sizes to be rejected and
generally one would expect a critical N of at least 200 for a good
model.
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The SPSS package for SEM – AMOS
• Graphical interface
• Alternative softwares
• LAMOS reads SPSS data sets
• ISREL, the first computer program which dates back to 1973 after
the pioneering work of the statistician Karl Jöreskog (1967 and
1969) which has evolved together with the method.
• SAS also provides a procedure for estimating structural equation
models proc CALIS.
• Other packages designed for structural equation models are EQS and
Mplus
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The example
• Behavioral model from the Trust data-set
• Theory of planned behaviour (Ajzen): human action
is determined by a combination of three
dimensions
• Behavioural beliefs (beliefs about the outcome of the
action) produce either a positive or a negative attitude
(A) toward the behaviour
• normative beliefs refer to subjective norms (SN) or
perceived social forces
• control beliefs lead to perceived behavioural control
(PBC)
• All these produce intentions to behave (ITB)
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Theory of planned behavior –
a path diagram
BB
A
RB
SN
CB
PBC
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ITB
30
Trust example
• The Trust data-set is based to some extent on the
theory of planned behaviour
• Intentions to purchase chicken (ITP) - question q7
• eleven behavioural beliefs (bba to bbk in the Trust
data-set) which act as the manifest indicators for the
latent construct attitude (A)
• For simplicity the subjective norm (SN) and perceived
behavioural control (PBC) are assumed to be measured
directly (although with error) through individual
measures PBC and SN in the Trust data-set
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Drawing the path diagram
• Open AMOS graphics from the AMOS menu
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The AMOS graphic interface
The path diagram can
be drawn directly here
These are elements and
functions to draw the
diagrams easily
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Some useful functions
Draw a latent variable or
adds indicators to a latent
variable
Draw a square
(manifest variable)
Draw a circle
(latent variable)
When clicking on an
indicator, associate it with a
latent unobserved variable
(generally an error)
Otherwise it creates the
latent variable
Draw a single arrow
(causation)
Draw a double arrow
(correlation)
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Copies
objects
Moves
objects
Delete
objects
34
The path diagram
1
1
1
1
1
1
1
1
1
1
1
1
1
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The data
• To include the observed variables it is now time to open
the SPSS file:
or simply click on this
button
Click on FILE NAME to open the
data, then click on OK
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Adding names of variables from the
data-set
Click here to see the list of
available variables
Simply drag the desired
variables on to the
desired square
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Unobservable variables and errors
• It is also necessary to give names to the latent variables and to the
errors (circles)
• Latent variables: Right-click on the desired circle, select PROPERTIES
and assign a a name and a label
• Errors: all remaining variables can be assigned an automated name as
follows
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The model is now ready
0,
e1
1
0,
e2
1
0,
e3
1
0,
e4
1
0,
e5
1
0,
e6
0,
e7
1
1
0,
e8
1
0,
0,
0,
e9 e10 e11
1
1
1
bba bbb bbc bbd bbe bbf bbg bbh bbi bbj bbk
1
0,
A
0,
e12
SN
1
ITP
PBC
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Before estimation
Intercept is required
for estimation if there
are missing data
Choose estimation
method
Decide whether results
for the independence
and saturated should
be shown
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Analysis properties
Set estimation
options
Ask for additional
output (e.g. factor
Convergenceweights for the
criteria
measurement model)
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Estimation
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Estimation output
As the estimation procedure
terminates this button becomes
available to show estimates
directly in the path diagram
This pane shows the final
Chi-square degrees of
freedom and signals
potential convergence
problems
As expected, the model is
over-identified
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Output view
Click here (or press F10)
to see the full output as
text
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AMOS output window
Summary information
Final estimates
Goodness-of-fit evaluation
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Degrees of freedom
and Chi-square
Variable summary
Parameter estimates
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Output
Computation of DoF(Default model)
Number of distinct sample moments: 119
Number of distinct parameters to be estimated:
45
Degrees of freedom (119 - 45):
74
DoF show that the
model is overidentified
The Chi-square statistic is high and
significant: this means that there is a
discrepancy between the observed
Result (Default model)
covariance matrix and the estimated one
Minimum was achieved
Chi-square = 619.428
Degrees of freedom = 74
Probability level = .000
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This is a frequent result but the Chi-square statistic
DoFtends to be inflated when there are many
DoF(or large sample sizes) and is very sensitive to
the assumption of multivariate normality
47
Estimates
Estimate
bba
<---
A
1.000
bbb
<---
A
.985
bbc
<---
A
bbd
<---
bbe
S.E.
C.R.
P
This coefficient is constrained to
one in order to ensure identification
.074
13.298
***
-.097
.067
-1.461
.144
A
.836
.067
12.496
***
<---
A
1.045
bbf
<---
A
.853
bbg
<---
A
.887
bbh
<---
A
.965
.079
12.296
***
bbi
<---
A
-.297
.064
-4.628
***
bbj
<---
A
.674
.077
8.807
***
bbk
<---
A
-.028
.084
-.327
.744
q7
<---
A
.073
.012
6.307
***
q7
<---
sn
.008
.008
1.051
.293
q7
<---
pbc
.016
.010
1.581
.114
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These
the loadings
of the
.078 are
13.485
***
manifest
indicators***on the latent
.061
13.872
variable
attitude
(A)
.074
11.953
***
Attitudes have a
positive and
significant (albeit
small)
impact
on
SN
& PBC
do not
intentions
have
a significant
impact on intentions
48
Standardized regression weights
Standardized weights can be interpreted like
correlations, but they assume causation
Estimate
bba
<---
A
.735
bbb
<---
A
.656
bbc
<---
A
-.072
bbd
<---
A
.616
bbe
<---
A
.661
bbf
<---
A
.682
bbg
<---
A
.594
bbh
<---
A
.633
bbi
<---
A
-.228
bbj
<---
A
.447
bbk
<---
A
-.017
q7
<---
A
.324
q7
<---
sn
.048
q7
<---
pbc
.071
The behavioral belief which is most important to
measure the latent construct (attitude) is bba, i.e.
chicken taste
These weights are negatively related to
attitudes. They measure the following beliefs:
bbc: difficulty of preparation
bbi: the agreement with the statement that
chicken lacks flavour
A standardized
weight
of 0.32
indicates a positive but
bbk: animal
welfare
concern
small relation between attitudes and intentions to
purchase chicken
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Correlations (no causation)
Covariances: (Group number 1 - Default model)
S.E.
C.R.
P
3.712
4.219
.880
.379
Estimate
sn
<-->
pbc
pbc
<-->
A
-2.880
3.548
-.812
.417
sn
<-->
A
12.017
4.461
2.694
.007
Correlations: (Group number 1 - Default model)
Estimate
sn
<-->
pbc
.040
pbc
<-->
A
-.040
sn
<-->
A
.134
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The only bidirectional relation which
emerges as significant is between
subjective norm (referent beliefs) and
attitudes
50
Goodness-of-fit (1)
Model Fit Summary
CMIN
Model
Default model
Saturated model
Independence model
NPAR
45
119
14
CMIN
619.428
.000
1736.822
DF
74
0
105
P
.000
CMIN/DF
8.371
.000
16.541
Baseline Comparisons Model
Model
Default model
Saturated model
Independence model
Parsimony-Adjusted Measures
Model
Default model
Saturated model
Independence model
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NFI
Delta1
RFI
rho1
.643
1.000
.000
.494
PRATIO
.705
.000
1.000
PNFI
.453
.000
.000
.000
IFI
Delta2
.672
1.000
.000
PCFI
.469
.000
.000
TLI
rho2
.526
.000
Values above fve
CFI
suggest
rejection
of the
.666 model
1.000
.000
In a good model all
these indicators should
be above 0.9
51
Goodness-of-fit (2)
NCP
Model
Default model
Saturated model
Independence model
FMIN
Model
Default model
Saturated model
Independence model
RMSEA
Model
Default model
Independence model
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NCP
545.428
.000
1631.822
LO 90
469.732
.000
1500.462
HI 90
628.594
.000
1770.572
FMIN
1.241
.000
3.481
F0
1.093
.000
3.270
LO 90
.941
.000
3.007
HI 90
1.260
.000
3.548
RMSEA
.122
.176
LO 90
.113
.169
HI 90
.130
.184
PCLOSE
.000
.000
Good models
have a pclose
value above
0.05
52
Information criteria
AIC
Model
Default model
Saturated model
Independence model
ECVI
Model
Default model
Saturated model
Independence model
HOELTER
Model
Default model
Independence model
AIC
709.428
238.000
1764.822
ECVI
1.422
.477
3.537
HOELTER
.05
77
38
BCC
712.218
245.376
1765.690
LO 90
1.270
.477
3.273
HI 90
1.588
.477
3.815
MECVI
1.427
.492
3.538
HOELTER
.01
85
41
Information and Hoelter criteria
are unsatisfactory
Statistics for Marketing & Consumer Research
Copyright © 2008 - Mario Mazzocchi
53
A competing model
• The theoretical model is rejected by the coefficients are
significant
• Competing model strategy
• try and remove all the non-significant components of the model
• add some other explanatory variables
• Problem: measurement of the latent construct for attitude
because the presence of items with negative wording might
lead to the identification of more than one latent factor
• Additional explanatory variable
• some variable explaining habit, which could be correlated with
attitude and influence ITP
• For example, variable q2b measures the frequency of chicken
purchases and we label it as habit
Statistics for Marketing & Consumer Research
Copyright © 2008 - Mario Mazzocchi
54
The modified model
0, 80.22
0, 89.38
0, 57.29
e4
e2
e1
0, 90.91
e5
1
1
1
bbb
1
29.38
34.27
bbd
32.42
bbe
1.08
37.31
.98
bba
.82
1.00
0, 71.04
14.54, 114.85
A
0, 2.76
SN
e12
.07
1
12.36
2.90
ITP
1.63, .93
Habits
Statistics for Marketing & Consumer Research
Copyright © 2008 - Mario Mazzocchi
3.85
.64
55
Estimates
Intercept
Intercept
Intercept
Intercept
Intercept
bba
bbb
bbd
bbe
q7
q7
Q2b
sn
<--<--<--<--<--<--<-->
<-->
bba
bbb
bbd
bbe
q7
Raw
Estimate
37.307
32.417
29.377
34.267
3.848
A
A
A
A
A
q2b
A
A
1.000
.975
.820
1.083
.066
.637
2.901
12.357
Statistics for Marketing & Consumer Research
Copyright © 2008 - Mario Mazzocchi
S.E.
.509
.567
.513
.593
.169
.080
.072
.086
.012
.091
.448
4.456
C.R.
73.344
57.216
57.300
57.774
22.763
12.131
11.426
12.647
5.490
7.022
6.479
2.773
P
Std.
Estimate
***
***
***
***
***
***
***
***
***
***
***
.006
.744
.656
.611
.692
.291
.319
.358
.137
56
Goodness-of-fit
Model
Final model
NPAR
22
CMIN
20.99
DF
13
P
0.073
CMIN/DF
1.615
NFI-Delta1
0.967
Final model
RFI-rho1
0.929
IFI-Delta2
0.987
TLI-rho2
0.972
CFI
0.987
PRATIO
0.464
PNFI
0.449
Final model
PCFI
0.458
NCP
7.99
LO 90
0
HI 90
24.616
FMIN
0.042
F0
0.016
Final model
LO 90
0
HI 90
0.049
RMS EA
0.035
LO 90
0
HI 90
0.062
PCLOS E
0.801
Final model
Saturated model
Independence model
AIC
64.99
70
652.525
BCC
65.707
71.141
652.753
ECVI
0.13
0.14
1.308
LO 90
0.114
0.14
1.15
HI 90
0.164
0.14
1.48
MECVI
0.132
0.143
1.308
Final model
Independence model
HOELTER
0.05
532
33
HOELTER
0.01
659
38
Statistics for Marketing & Consumer Research
Copyright © 2008 - Mario Mazzocchi
The diagnostics are now ok
57