Transcript Chapter One

Chapter Twenty-Two
Structural Equation
Modeling and Path
Analysis
Copyright © 2010 Pearson Education, Inc.
22-1
Chapter Outline
1)
Objectives
2)
Overview
3)
Basic Concepts in SEM
i.
Theory, Model and Path Diagram
ii.
Exogenous versus Endogenous Constructs
iii.
Dependence and Correlational Relationships
iv.
Model Fit
Model Identification
4)
Statistics Associated with SEM
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Chapter Outline
5)
6)
7)
i.
8)
Conducting SEM
Define the Individual Constructs
Specify The Measurement Model
Sample Size Requirements
Assess Measurement Model Reliability and
Validity
i.
Assess Measurement Model Fit
a.
b.
c.
d.
e.
chi-square (χ2)
Absolute Fit Indices: Goodness-of-Fit
Absolute Fit Indices: Badness-of-Fit
Incremental Fit Indices
Parsimony Fit Indices
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Chapter Outline
ii. Measurement Model Reliability and Validity
a.
Reliability
b.
9)
10)
Discriminant Validity
iii. Lack of Validity: Diagnosing Problems
Specify the Structural Model
Assess Structural Model Validity
i. Assessing Fit
ii. Comparison with Competing Models
iii. Testing Hypothesized Relationships
iv. Structural Model Diagnostics
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Chapter Outline
11)
12)
13)
14)
15)
16)
17)
Draw Conclusions and Make
Recommendations
Higher-Order Confirmatory Factor Analysis
Relationship of SEM to Other Multivariate
Technique
Application of SEM: First-Order Factor Model
Application of SEM: Second-Order Factor Model
Path Analysis
Statistical Software
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Structural Equation Modeling (SEM)
Structural equation modeling (SEM), a
procedure for estimating a series of dependence
relationships among a set of concepts or
constructs represented by multiple measured
variables and incorporated into an integrated
model.
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Structural Equation Modeling: Distinctive Aspects
1. Representation of constructs as unobservable or
latent factors in dependence relationships.
2. Estimation of multiple and interrelated dependence
relationships incorporated in an integrated model.
3. Incorporation of measurement error in an explicit
manner. SEM can explicitly account for less than
perfect reliability of the observed variables,
providing analyses of attenuation and estimation
bias due to measurement error.
4. Explanation of the covariance among the observed
variables.
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Statistics Associated with SEM
Absolute fit indices These indices measure the overall
goodness-of-fit or badness-of-fit for both the
measurement and structural models.
Average variance extracted A measure used to
assess convergent and discriminant validity, which is
defined as the variance in the indicators or observed
variables that is explained by the latent construct.
Chi-square difference statistic ( Δχ2) A statistic used
to compare two competing, nested SEM models. It is
calculated as the difference between the models’ chisquare value. Its degrees of freedom equal the difference
in the models’ degrees of freedom.
Communality Communality is the variance of a
measured variable that is explained by its construct.
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Statistics Associated with SEM
Composite reliability (CR) It is defined as the
total amount of true score variance in relation to the
total score variance.
Confirmatory factor analysis (CFA) A technique
used to estimate the measurement model. It seeks to
confirm if the number of factors (or constructs) and
the loadings of observed (indicator) variables on them
conform to what is expected on the basis of theory.
Construct In SEM, a construct is a latent or
unobservable concept that can be defined
conceptually but that cannot be measured directly or
without error. Also called a factor, a construct is
measured by multiple indicators or observed
variables.
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Statistics Associated with SEM
Endogenous constructs An endogenous construct is
the latent, multi-item equivalent of a dependent variable.
It is determined by constructs or variables within the
model and, thus, it is dependent on other constructs.
Estimated covariance matrix Denoted by Σk , it
consists of the predicted covariances between all
observed variables based on equations estimated in SEM.
Exogenous construct An exogenous construct is the
latent, multi-item equivalent of an independent variable
in traditional multivariate analysis. An exogenous
construct is determined by factors outside of the model
and it cannot be explained by any other construct or
variable in the model.
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Statistics Associated with SEM
First-order factor model Covariances between
observed variables are explained with a single latent
factor or construct layer.
Incremental fit indices These measures assess
how well a model specified by the researcher fits
relative to some alternative baseline model.
Typically, the baseline model is a null model in
which all observed variables are unrelated to each
other.
Measurement error It is the degree to which the
observed variables do not describe the latent
constructs of interest in SEM.
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Statistics Associated with SEM
Measurement model The first of two models
estimated in SEM. It represents the theory that
specifies the observed variables for each construct
and permits the assessment of construct validity.
Modification index An index calculated for each
possible relationship that is not freely estimated but
is fixed. The index shows the improvement in the
overall model χ2 if that path was freely estimated.
Nested model A model is nested within another
model if it has the same number of constructs and
variables and can be derived from the other model
by altering relationships, as by adding or deleting
relationships.
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Statistics Associated with SEM
Nonrecursive model A structural model that
contains feedback loops or dual dependencies.
Parsimony fit indices The parsimony fit indices
are designed to assess fit in relation to model
complexity and are useful in evaluating competing
models. These are goodness-of-fit measures and
can be improved by a better fit or by a simpler, less
complex model that estimates fewer parameters.
Parsimony ratio Is calculated as the ratio of
degrees of freedom used by the model to the total
degrees of freedom available.
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Statistics Associated with SEM
Path analysis A special case of SEM with only single
indicators for each of the variables in the causal model.
In other words, path analysis is SEM with a structural
model, but no measurement model.
Path diagram A graphical representation of a model
showing the complete set of relationships amongst the
constructs. Dependence relationships are portrayed by
straight arrows and correlational relationships by curved
arrows.
Residuals In SEM, the residuals are the differences
between the observed and estimated covariance
matrices.
Recursive model A structural model that does not
contain any feedback loops or dual dependencies.
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Statistics Associated with SEM
Sample covariance matrix Denoted by S, it
consists of the variances and covariances for the
observed variables.
Second-order factor model There are two
levels or layers. A second-order latent construct
causes multiple first-order latent constructs, which
in turn cause the observed variables. Thus, the
first-order constructs now act as indicators or
observed variables for the second order factor.
Squared multiple correlations Similar to
communality, these values denote the extent to
which an observed variable’s variance is explained
by a latent construct or factor.
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Statistics Associated with SEM
Standardized residuals Used as a diagnostic
measure of model fit, these are residuals, each
divided by its standard error.
Structural error Structural error is the same as
an error term in regression analysis. In the case of
completely standardized estimates, squared
multiple correlation is equal to 1 – the structural
error.
Structural model The second of two models
estimated in SEM. It represents the theory that
specifies how the constructs are related to each
other, often with multiple dependence
relationships.
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Statistics Associated with SEM
Structural relationship Dependence relationship
between an endogenous construct and another
exogenous or endogenous construct.
Unidimensionality A notion that a set of
observed variables represent only one underlying
construct. All cross-loadings are zero.
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Exogenous and Endogenous Constructs
Exogenous constructs are the latent, multi-item equivalent
of independent variables. They use a variate (linear
combination) of measures to represent the construct, which
acts as an independent variable in the model.
Multiple measured variables (X) represent the exogenous
constructs (ξ).
Endogenous constructs are the latent, multi-item equivalent
to dependent variables. These constructs are theoretically
determined by factors within the model.
Multiple measured variables (Y) represent the endogenous
constructs (η).
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SEM Models
Models can be represented visually with a path diagram.
Dependence relationships are represented with singleheaded straight arrows.
Correlational (covariance) relationships are
represented with two-headed curved arrows.
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Dependence and Correlational Relationships in SEM
Fig. 22.1
(a) Dependence Relationship
Endogenous
Construct: C2
Exogenous
Construct: C1
X1
X2
X3
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Y1
Y2
Y3
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Dependence and Correlational Relationships in SEM
Fig. 22.1 Cont.
Exogenous
Construct: C1
X1
X2
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Exogenous
Construct: C2
X3
X4
X5
X6
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Conducting SEM
Steps in SEM
Step
Step
Step
Step
Step
Step
1:
2:
3:
4:
5:
6:
Define the Individual Constructs
Specify the Measurement Model
Assess Measurement Model Reliability and Validity
Specify the Structural Model
Assess Structural Model Validity
Draw Conclusions and Make Recommendations
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The Process for Structural Equation Modeling
Define the Individual Constructs
Fig. 22.2
Develop and Specify the Measurement Model
Assess Measurement Model Reliability and Validity
Refine Measures and
Design a New Study
NO
Measurement Model
Valid?
YES
Specify the Structural Model
Assess Structural Model Validity
Refine Model and
Test with New
Data
No
Structural Model Valid?
YES
Draw Conclusions and Make Recommendations
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Path Diagram of a Simple Measurement Model
Fig. 22.3
Φ21
C2
ξ2
C1
ξ1
λ
λ
λх3,1
x1,1
λ
X4,2
X2,1
λ
λ
X5,2
x6,2
X1
X2
X3
X4
X5
X6
δ1
δ2
δ3
δ4
δ5
δ6
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A Simple Measurement Model
In Figure 22.3, ξ1 represents the latent construct C1 ,
ξ2 represents the latent construct C2, x1 - x6 represent
the measured variables, λx1,1 - λχ6,2 represent the
relationships between the latent constructs and the
respective measured items (i.e; factor loadings), and
δ1 - δ6 represent the errors.
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Types of Fit Measures
1. Absolute fit measures overall goodness- or badness-of-fit for
both the structural and measurement models. This type of
measure does not make any comparison to a specified null
model (incremental fit measure) or adjusts for the number of
parameters in the estimated model (parsimonious fit
measure).
2. Incremental fit measures goodness-of-fit that compares the
current model to a specified “null” (independence) model to
determine the degree of improvement over the null model.
3. Parsimonious fit measures goodness-of-fit representing the
degree of model fit per estimated coefficient. This measure
attempts to correct for any “overfitting” of the model and
evaluates the parsimony of the model compared to the
goodness-of-fit.
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A Classification of Fit Measures
Fit Measures
Fig. 22.4
Absolute Fit
Indices
Goodness-of-Fit
Badness-of-Fit
Incremental
Fit Indices
Goodness-of-Fit
Parsimony
Fit Indices
Goodness-of-Fit
• GFI
• x2
• NFI

PGFI
• AGFI
• RMSR
• NNFI

PNFI
• SRMR
• CFI
• RMSEA
• TLI
• RNI
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Assessing Model Fit: Chi-square
  (n 1)(S  k )

1
df  [( p )( p  1)]  k
2
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Assessing Model Fit: Fit Indexes
Multiple fit indices should be used to assess a model’s
goodness of fit. They should include:
•
The χ2 value and the associated df
•
Two absolute fit indices (GFI, AGFI, RMSEA, or SRMR)
•
•
•
One goodness-of-fit index (GFI, AGFI)
•
One badness-of-fit index (RMSR, SRMR, RMSEA)
One incremental fit index (CFI, TLI, NFI, NNFI, RNI)
One parsimony fit index for models of different
complexities (PGFI, PNFI)
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Composite Reliability (CR)
CR =
p
(∑ λi )2
i=1
p
p
(∑λi)2 + (∑δi)
i=1
i=1
Where
CR
=
Composite reliability
λ
=
completely standardized factor loading
δ
=
Error variance
p
=
number of indicators or observed variables
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Average Variance Extracted (AVE)
p 2
Σ λί
i=1
AVE =
p 2
Σ λί +
i=1
p
Σ δi
i=1
Where
AVE
=
Average variance extracted
λ
=
Completely Standardized factor loading
δ
=
Error variance
p
=
number of indicators or observed variables
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Path Diagram of a Simple Structural Model
Fig. 22.5
γ1,1
Construct 1: C1
ξ1
λx1,1
λx3,1
λx2,1
X1
δ1
X2
δ2
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X3
δ3
Construct 2:C2
η1
λy3,1
λy1,1
λy2,1
Y1
ε1
Y2
ε2
Y3
ε3
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Chi-Square Difference Test
Δχ2Δdf = χ2df(M1) - χ2df(M2)
and
Δdf
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= df(M1) –df(M2)
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First-Order Model of IUIPC
Fig. 22.6
Ф3,1
Φ2,1
Ф3,2
COL
X1
X2
X3
CON
X4
X5
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X6
AWA
X7
X8
X9
X10
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Second-Order Model of IUIPC
Fig. 22.7
X1,1
Layer 1
X2
X3,1
X2,1
COL
X1
Layer 2
IUIPC
CON
X3
X4
X5
X6
AWA
X7
X8
X9
X10
Legend:
First-Order Factors:
COL = Collection
CON = Control
AWA = Awareness
Second-Order Factor :
IUIPC = Internet Users Information Privacy Concerns
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Measurement Model for TAM
Fig. 22.8
Ф3,1
Φ2,1
Ф3,2
PU
PU1 PU2
PU3
PE
PE1
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PE2
INT
PE3
INT1
INT2
INT3
22-36
Structural Model for TAM
Fig. 22.9
Perceived
Usefulness
PU
0.60
0.46
Intention
to Use
INT
0.40
Perceived
Ease Of Use
PE
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0.28
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TAM: Means, Standard Deviations, and Correlations
Table 22.1
1. PU1
2. PU2
3. PU3
4. PE1
5. PE2
6. PE3
7. INT1
8. INT2
9. INT3
ME
SD
3.58
3.58
3.58
4.70
4.76
4.79
3.72
3.84
3.68
1.37
1.37
1.36
1.35
1.34
1.32
2.10
2.12
2.08
Correlations Matrix
1
2
3
4
5
6
7
8
9
1
.900**
.886**
.357**
.350**
.340**
.520**
.513**
.534**
1
.941**
.403**
.374**
.356**
.545**
.537**
.557**
1
.392**
.393**
.348**
.532**
.540**
.559**
1
.845**
.846**
.442**
.456**
.461**
1
.926**
.419**
.433**
.448**
1
.425**
.432**
.437**
1
.958**
.959**
1
.950**
1
Notes:
 ME = means; SD = standard deviations;
 *p<0.05, **p<0.01, ***p<0.001 (two-tailed).
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TAM: Results of Measurement Model
Table 22.2
Constructs
PU
Items
Item Loadings
Item Errors
PU1
PU2
PU3
0.92***
0.98***
0.96***
0.15***
0.05***
0.07***
PE1
PE2
PE3
0.88***
0.23***
0.96***
0.96***
0.07***
0.08***
INT2
INT2
INT3
0.98***
0.03***
0.97***
0.98***
0.05***
0.05***
PE
INT
CR
0.97
AVE
0.91
0.95
0.87
0.98
0.95
Notes:
 CR = composite reliability; AVE = average variance extracted.
 *p<0.05, **p<0.01, ***p<0.001 (two-tailed).
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Banking Application: Measurement Model
Fig. 22.10
TANG ξ1
x1
x2
δ1 δ2
x3
REL ξ2
x4
δ3 δ4
X5
δ5
X6
δ6
X7
δ7
RESP ξ3
X8
δ8
X9
δ9
x10
δ10
X11
δ11
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ASS4 ξ4
x12
δ12
x13
x14
X15
δ13 δ14 δ15
EMP ξ5
X16
X17
x18
x19
δ16 δ17 δ18 δ19
ATT ξ6
x20
x21
x22
SAT ξ7
x23
x24
x25
PAT ξ8
x26
x27 x28
X29
δ20 δ21 δ22 δ23 δ24 δ25 δ26 δ27 δ28 δ29δ30
22-40
x30
Banking Application: Structure Model
Fig. 22.11
TANG
Service
Attitude
REL
RESP
Service
Quality
Patronage
Intention
ASSU
Service
Satisfaction
EMP
TANG = tangibility; REL= reliability; RESP = responsiveness; ASSU = assurance; EMP = empathy
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Psychometric Properties of Measurement Model
Table 22.3
When it comes to….
My Perception of My
Bank’s Service.
Loadings
TANG1: Modern equipments
TANG2: Visual appeal of physical facilities
TANG3: Neat, professional appearance of employees
TANG4: Visual appeal of materials associated with the service
REL1: Keeping a promise by a certain time
REL2: Performing service right the first time
REL3: Providing service at the promised time
REL4: Telling customers the exact time the service will be performed
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0.71
0.80
0.76
0.72
0.79
0.83
0.91
0.81
Measurement Error
0.49
0.36
0.42
0.48
0.37
0.31
0.18
0.34
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Psychometric Properties of Measurement Model
Table 22.3 Cont.
When it comes to….
My Perception of My
Bank’s Service.
Loadings
RESP1: Keeping a promise by a certain time
RESP2: Willingness to always help customers
RESP3: Responding to customer requests despite being busy
ASSU1: Employees instilling confidence in customers
ASSU2: Customers’ safety feelings in transactions
(e.g. physical, financial, emotional, etc.)
ASSU3: Consistent courtesy to customers
ASSU4: Employees’ knowledge to answer customer questions
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Measurement Error
0.73
0.89
0.81
0.81
0.71
0.47
0.21
0.35
0.35
0.49
0.80
0.86
0.36
0.26
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Psychometric Properties of Measurement Model
Table 22.3 Cont.
When it comes to….
My Perception of My
Bank’s Service.
Loadings
EMP1: Giving customers individual attention
EMP2: Dealing with customers with care
EMP3: Having customers’ best interests at heart
EMP4: Understanding specific needs of customers
Overall attitude toward your bank (items reverse coded):
ATT1: Favorable 1---2---3---4---5---6---7 Unfavorable
ATT2: Good 1---2---3---4---5---6---7 Bad
ATT3: Positive 1---2---3---4---5---6---7 Negative
ATT4: Pleasant 1---2---3---4---5---6---7 Unpleasant
SAT1: I believe I am satisfied with my bank’s services
SAT2: Overall, I am pleased with my bank’s services
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Measurement Error
0.80
0.84
0.87
0.87
0.37
0.29
0.24
0.24
0.95
0.95
0.95
0.95
0.93
0.93
0.10
0.10
0.10
0.10
0.14
0.14
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Psychometric Properties of Measurement Model
Table 22.3 Cont.
When it comes to….
My Perception of My
Bank’s Service.
Loadings
SAT3: Using services from my bank is usually a satisfying
experience
SAT4: My feelings toward my bank’s services can best be
characterized as
PAT1: The next time my friend needs the services of a bank
I will recommend my bank
PAT2: I have no regrets of having patronized my bank in the past
PAT3: I will continue to patronize the services of my bank in the
future
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Measurement Error
0.88
0.23
0.92
0.15
0.88
0.22
0.89
0.88
0.20
0.22
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Goodness of Fit Statistics Measurement Model
Table 22.4
---------------------------------------------Degrees of Freedom = 377
Minimum Fit Function Chi-Square =
767.77 (P = 0.0)
Chi-Square for Independence Model with
435 Degrees of Freedom = 7780.15
Root Mean Square Error of Approximation
(RMSEA) = 0.064
Standardized RMR = 0.041
Normed Fit Index (NFI) = 0.90
Non-Normed Fit Index (NNFI) = 0.94
Comparative Fit Index (CFI) = 0.95
---------------------------------------------Copyright © 2010 Pearson Education, Inc.
22-46
Measurement Model: Construct Reliability,
Average Variance Extracted & Correlation Matrix
Table 22.5
Construct
Construct
Reliability
1. TANG
Average
Variance
Extracted
Correlation Matrix
1
2
3
4
0.56
0.75
5
6
7
8
0.84
2. REL
0.90
0.70
0.77
0.84
3. RESP
0.85
0.66
0.65
0.76 0.81
4. ASSU
0.87
0.63
0.73
0.80 0.92 0.80
5. EMP
0.91
0.71
0.69
0.75 0.85 0.90 0.85
6. ATT
0.97
0.90
0.42
0.46 0.52 0.54 0.58 0.95
7. SAT
0.85
0.83
0.53
0.56 0.66 0.67 0.69 0.72 0.91
8. PAT
0.92
0.78
0.50
0.55 0.57 0.62 0.62 0.66 0.89 0.89
TANG = tangibles; REL = reliability; RESP = responsiveness; ASSU = assurance; EMP = empathy; ATT = attitude;
SAT = satisfaction; PAT = patronage.
Value on the diagonal of the correlation matrix is the square root of AVE.
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Goodness of Fit Statistics (Structural Model)
Table 22.6
--------------------------------------------------------------------------------Degrees of Freedom = 396
Minimum Fit Function Chi-Square = 817.16 (P = 0.0)
Chi-Square for Independence Model with 435 Degrees of Freedom = 7780.15
Root Mean Square Error of Approximation (RMSEA) = 0.065
Standardized RMR = 0.096
Normed Fit Index (NFI) = 0.89
Non-Normed Fit Index (NNFI) = 0.94
Comparative Fit Index (CFI) = 0.94
--------------------------------------------------------------------------------------------------------------Copyright © 2010 Pearson Education, Inc.
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Banking Application: Structural Coefficients
Table 22.7
Second Order
Loading Estimates
Dimensions of
Service Quality
T-values
TANG
g11
0.82
l fixed to 1
REL
g1
0.85
13.15
RESP
g31
0.93
13.37
ASSU
g41
0.98
16.45
EMP
g51
0.93
15.18
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Banking Application: Structural Coefficients
Table 22.7 Cont.
Structural
Coefficient
Estimates
Consequences of
Service Quality
T-Values
SQATT
g61
0.60
10.25
SQSAT
g71
0.45
8.25
ATTSAT
b76
0.47
8.91
ATTPAT
b86
0.03
0.48
SATPAT
b87
0.88
13.75
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22-50
Path Analysis
Path analysis A special case of SEM
with only single indicators for each of the
variables in the causal model. In other
words, path analysis is SEM with a
structural model, but no measurement
model.
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22-51
Diagram for Path Analysis
Fig. 22.12
X1
Attitude
B
Y1
Purchase
Intention
A
X2
Emotion
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C
22-52
Path Analysis: Calculating Structural Coefficients
Figure 22.12 portrays a simple model with two exogenous constructs X1 and
X2 causally related to the endogenous construct Y1. The correlational path A
is X1 correlated with X2. Path B is the effect of X1 predicting Y1, and path C
shows the effect of X2 predicting Y1. The value for Y1 can be modeled as:
Y1 = b1 X1 + b2X2
Note that this is similar to a regression equation. The direct and indirect
paths in our model can now be identified.
Direct Paths
A=X1 to X2
B=X1 to Y1
C=X2 to Y1
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Indirect Paths
AC = X1 to Y1 (Via X2)
AB = X2 to Y1 (Via X1)
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Path Analysis: Calculating Structural Coefficients
The unique correlations among the three constructs can be shown to be
composed of direct and indirect paths as follows:
Corrx1, x2 = A
Corrx1, y1 = B + AC
Corrx2, y1 = C + AB
The correlation of X1 and X2 is simply equal to A. The Correlation
of X1 and Y1 ( Corrx1,y1 ) can be represented by two paths:B and AC.
B represents the direct path from X1 to Y1. AC is a compound path
that follows the curved arrow from X1 to X2 and then to Y1. Similarly, the
correlation of X2 and Y1 can be shown to consist of two causal paths: C and AB.
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Path Analysis
Bivariate Correlations
X1
X1
X2
Y1
1.0
X2
.40
Y1
.50
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1.0
.60
1.0
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Path Analysis
Correlations as Compound Paths
Corr
X1, X2
=A
Corr
X1 ,Y1
= B+AC
Corr
X2 ,Y1
= C+AB
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Path Analysis: Calculating Structural Coefficients
.40 = A
.50 = B+AC
.60 = C+AB
Substituting A = .40
.50 = B+.40C
.60 = C+.40B
Solving for B and C
B = .310
C = .476
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Statistical Software
•LISREL, LInear Structural RELations
•AMOS, Analysis of Moment Structures, is an added
module to SPSS
•CALIS is offered by SAS
•EQS, an abbreviation for equations
•Mplus is another software
Selection of a specific computer program should be based
on availability and user’s preference.
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Copyright © 2010 Pearson Education, Inc.
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reproduced, stored in a retrieval system, or transmitted, in
any form or by any means, electronic, mechanical,
photocopying, recording, or otherwise, without the prior
written permission of the publisher. Printed in the United
States of America.
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Copyright © 2010 Pearson Education, Inc.
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