copyright © 2010 pearson education, inc. 22-1 chapter twenty-two structural equation modeling and...

Copyright © 2010 Pearson Education, Inc. 22-1

Chapter Twenty-Two

Structural Equation Modeling and Path Analysis


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


Chapter Outline

5) Conducting SEM6) Define the Individual Constructs7) Specify The Measurement Model

i. Sample Size Requirements8) Assess Measurement Model Reliability and Validity

i. Assess Measurement Model Fita. chi-square (χ2)

b. Absolute Fit Indices: Goodness-of-Fitc. Absolute Fit Indices: Badness-of-Fitd. Incremental Fit Indicese. Parsimony Fit Indices


Chapter Outline

ii. Measurement Model Reliability and Validity

a. Reliability

b. Discriminant Validity

iii. Lack of Validity: Diagnosing Problems9) Specify the Structural Model 10) Assess Structural Model Validity i. Assessing Fit

ii. Comparison with Competing Models

iii. Testing Hypothesized Relationships

iv. Structural Model Diagnostics


Chapter Outline

11) Draw Conclusions and Make Recommendations

12)Higher-Order Confirmatory Factor Analysis13)Relationship of SEM to Other Multivariate

Technique14)Application of SEM: First-Order Factor Model15)Application of SEM: Second-Order Factor Model16)Path Analysis17)Statistical Software


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.

Structural Equation Modeling (SEM)


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.

Structural Equation Modeling: Distinctive Aspects


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’ chi-square 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.

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.



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.



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.



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.



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.



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.



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.



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.



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.



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).


Models can be represented visually with a path diagram.

Dependence relationships are represented with single-headed straight arrows.

Correlational (covariance) relationships are represented with two-headed curved arrows.

SEM Models


(a) Dependence Relationship

ExogenousConstruct: C1

EndogenousConstruct: C2

X1 X2 X3Y1 Y2 Y3

Dependence and Correlational Relationships in SEM

Fig. 22.1Fig. 22.1Fig. 22.1




X1 X2 X3 X4X5 X6

Dependence and Correlational Relationships in SEM

Fig. 22.1 Cont.


Conducting SEM

Steps in SEM

Step 1: Define the Individual Constructs Step 2: Specify the Measurement ModelStep 3: Assess Measurement Model Reliability and

ValidityStep 4: Specify the Structural ModelStep 5: Assess Structural Model ValidityStep 6: Draw Conclusions and Make

Recommendations


The Process for Structural Equation Modeling

Define the Individual Constructs

Develop and Specify the Measurement Model

Assess Measurement Model Reliability and Validity

Specify the Structural Model

Assess Structural Model Validity

Draw Conclusions and Make Recommendations

Measurement ModelValid?

Refine Measures andDesign a New Study

YES

NO

Structural Model Valid?

YES

Refine Model and Test with New

Data

No

Fig. 22.2


Path Diagram of a Simple Measurement Model

C1

ξ1

C2

ξ2

X1 X2 X3 X4 X5 X6

λx1,1 λ

X2,1

λX4,2

λX5,2

λx6,2

Φ21

δ4 δ5 δ6 δ1 δ2 δ3

λх3,1

Fig. 22.3


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.

A Simple Measurement Model


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.

Types of Fit Measures


A Classification of Fit Measures

Absolute Fit Incremental Parsimony Indices Fit Indices Fit Indices

Goodness-of-Fit Badness-of-Fit Goodness-of-Fit

• NFI

• NNFI

• CFI

• TLI

• RNI

Goodness-of-Fit

PGFI

PNFI

• GFI

• AGFI

• x2

• RMSR

• SRMR

• RMSEA

Fit MeasuresFig. 22.4


Assessing Model Fit: Chi-square

))(1( kSn

kppdf )]1)([(2

1


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)

Assessing Model Fit: Fit Indexes


Where

CR = Composite reliability

λ = completely standardized factor loading

δ = Error variance

p = number of indicators or observed variables

CR = (∑ λi )2

(∑λi)2 + (∑δi)

p

i=1

i=1 i=1

p p

Composite Reliability (CR)


p 2 Σ λί

i=1 AVE = p 2 p

Σ λί + Σ δi

i=1 i=1

Where

AVE = Average variance extracted

λ = Completely Standardized factor loading

δ = Error variance

p = number of indicators or observed variables

Average Variance Extracted (AVE)


Path Diagram of a Simple Structural Model

Construct 1: C1

ξ1

Construct 2:C2 η1

X1 X2 X3 Y1 Y2 Y3

γ1,1

λx1,1

λx2,1

λy1,1

λy2,1

δ1 δ2δ3

ε1 ε2 ε3

λy3,1λx3,1

Fig. 22.5


Δχ2Δdf = χ2

df(M1) - χ2df(M2)

and

Δdf = df(M1) –df(M2)

Chi-Square Difference Test


First-Order Model of IUIPC

COL CON AWA

X1 X2 X3 X4 X5 X6X7 X8 X9 X10

Φ2,1 Ф3,2

Ф3,1

Fig. 22.6


Second-Order Model of IUIPC

IUIPC Layer 2

COL CON AWA

Layer 1

X1 X3 X4X2 X5 X6 X7 X10X8 X9

X1,1 X2,1

X3,1

Legend:First-Order Factors: COL = Collection CON = Control AWA = Awareness

Second-Order Factor : IUIPC = Internet Users Information Privacy Concerns

Fig. 22.7


Measurement Model for TAM

PU PE INT

PU1 PU2 PU3 PE1 PE2 PE3 INT1 INT2 INT3

Φ2,1 Ф3,2

Ф3,1Fig. 22.8


Structural Model for TAM

Perceived Usefulness

PU

Perceived Ease Of Use

PE

Intentionto Use

INT

0.46

0.28

0.40

0.60

Fig. 22.9


ME SD Correlations Matrix

1 2 3 4 5 6 7 8 9

1. PU1 3.58 1.37 1 2. PU2 3.58 1.37 .900** 1 3. PU3 3.58 1.36 .886** .941** 1 4. PE1 4.70 1.35 .357** .403** .392** 1 5. PE2 4.76 1.34 .350** .374** .393** .845** 1 6. PE3 4.79 1.32 .340** .356** .348** .846** .926** 1 7. INT1 3.72 2.10 .520** .545** .532** .442** .419** .425** 1 8. INT2 3.84 2.12 .513** .537** .540** .456** .433** .432** .958** 1 9. INT3 3.68 2.08 .534** .557** .559** .461** .448** .437** .959** .950** 1

Notes: ME = means; SD = standard deviations; *p<0.05, **p<0.01, ***p<0.001 (two-tailed).

TAM: Means, Standard Deviations, and CorrelationsTable 22.1Table 22.1


Constructs Items Item Loadings Item Errors CR AVE

PU 0.97 0.91 PU1 0.92*** 0.15*** PU2 0.98*** 0.05*** PU3 0.96*** 0.07***

PE 0.95 0.87 PE1 0.88*** 0.23*** PE2 0.96*** 0.07*** PE3 0.96*** 0.08***

INT 0.98 0.95 INT2 0.98*** 0.03*** INT2 0.97*** 0.05*** INT3 0.98*** 0.05***

Notes: CR = composite reliability; AVE = average variance extracted. *p<0.05, **p<0.01, ***p<0.001 (two-tailed).

TAM: Results of Measurement Model

Table 22.2


x1 x2x3

x4X5

X6 X7X8

X9 x10 X11 x12 x13 x14 X16X15 X17 x18 x19 x20 x21 x22 x23x28

x24 X29

TANG ξ1 REL ξ2 RESP ξ3 ASS4 ξ4 EMP ξ5 ATT ξ6 SAT ξ7 PAT ξ8

δ1 δ2 δ3 δ4 δ5 δ6 δ7 δ8 δ9 δ10 δ11 δ12 δ13 δ14 δ15 δ16 δ17 δ18 δ19 δ20 δ21 δ22 δ23 δ24 δ25 δ26 δ27 δ28 δ29 δ30

Banking Application: Measurement Model

x25 x26 x27 x30

Fig. 22.10


Service Quality

TANG

REL

RESP

ASSU

Patronage Intention

Service Attitude

Service Satisfaction

EMP

TANG = tangibility; REL= reliability; RESP = responsiveness; ASSU = assurance; EMP = empathy

Banking Application: Structure Model

Fig. 22.11


When it comes to…. My Perception of My

Bank’s Service.

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

Loadings Measurement Error

0.71 0.49

0.80 0.36

0.76 0.42

0.72 0.48

0.79 0.37

0.83 0.31

0.91 0.18

0.81 0.34

Psychometric Properties of Measurement ModelTable 22.3



Bank’s Service.

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


0.73 0.47

0.89 0.21

0.81 0.35

0.81 0.35

0.71 0.49

0.80 0.36

0.86 0.26

Psychometric Properties of Measurement Model

Table 22.3 Cont.



Bank’s Service.

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


0.80 0.37

0.84 0.29

0.87 0.24

0.87 0.24

0.95 0.10

0.95 0.10

0.95 0.10

0.95 0.10

0.93 0.14

0.93 0.14


Table 22.3 Cont.



Bank’s Service.

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


0.88 0.23

0.92 0.15

0.88 0.22

0.89 0.20

0.88 0.22

Table 22.3 Cont.



----------------------------------------------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) = 0.90Non-Normed Fit Index (NNFI) = 0.94Comparative Fit Index (CFI) = 0.95

----------------------------------------------

Table 22.4

Goodness of Fit Statistics Measurement Model


Measurement Model: Construct Reliability, Average Variance Extracted & Correlation Matrix

Construct ConstructReliability

AverageVarianceExtracted

Correlation Matrix1 2 3 4 5 6 7 8

1. TANG

0.84

0.56 0.75

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.

Table 22.5


---------------------------------------------------------------------------------

Degrees of Freedom = 396Minimum Fit Function Chi-Square = 817.16 (P = 0.0)

Chi-Square for Independence Model with 435 Degrees of Freedom = 7780.15Root Mean Square Error of Approximation (RMSEA) = 0.065

Standardized RMR = 0.096

Normed Fit Index (NFI) = 0.89Non-Normed Fit Index (NNFI) = 0.94Comparative Fit Index (CFI) = 0.94

---------------------------------------------------------------------------------------------------------------

Table 22.6

Goodness of Fit Statistics (Structural Model)


Banking Application: Structural Coefficients

Dimensions of Service Quality

Second Order T-values Loading Estimates

TANG 0.82 fixed to 1

REL 0.85 13.15

RESP 0.93 13.37

ASSU 0.98 16.45

EMP 0.93 15.18

Table 22.7Table 22.7


SQATT 0.60 10.25

SQSAT 0.45 8.25

ATTSAT 0.47 8.91

ATTPAT 0.03 0.48

SATPAT 0.88 13.75

Consequences of

Service Quality

StructuralCoefficient Estimates

T-Values

Table 22.7 Cont.

Banking Application: Structural Coefficients


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 Analysis


Diagram for Path Analysis

B

C

A

X1

Attitude

X2

Emotion

Y1

PurchaseIntention

Fig. 22.12


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 Indirect PathsA=X1 to X2

B=X1 to Y1

C=X2 to Y1

AC = X1 to Y1 (Via X2)AB = X2 to Y1 (Via X1)

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 + ACCorrx2, y1 = C + AB

The correlation of X1 and X2 is simply equal to A. The Correlationof 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.


Bivariate Correlations

X1 X2 Y1

X1 1.0

X2 .40 1.0

Y1 .50 .60 1.0

Path Analysis


Correlations as Compound Paths

Corr X1, X2 = A

Corr X1 ,Y1 = B+AC

Corr X2 ,Y1 = C+AB

Path Analysis


.40 = A

.50 = B+AC

.60 = C+AB

Substituting A = .40.50 = B+.40C.60 = C+.40B

Solving for B and CB = .310C = .476



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.

copyright © 2010 pearson education, inc. 22-1 chapter twenty-two structural equation modeling and...

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