structural eqqguation modeling - iacmriacmr.org/v2/chineseweb/main/sem (lecture 7) mar 28 am.pdf ·...

Structural Equation Modelingq g1

Correlation and Regression Covariance Structure Analysis, Structural

Equation Modeling and LISRELEquation Modeling and LISREL What is SEM? How are estimations obtained? The idea of goodness-of-fit Model Identification Testing nested models Testing nested models

Kenneth Law @ 同济大学 2010

The idea of CFA2

Confirmatory Factor AnalysisConfirmatory Factor Analysis

(Exploratory Factor Analysis) EFA


Two measurement modelsTwo measurement models

Fairness Classical measurement model

3

Fairnessperception x1 = 1 + 1

x2 = 1 + 2

Latent

2 1 2

C i t d l

x1 = 11 1 + 1

Item2Item1 x1 x2

Latent Co-generic measurement model

x2 = 21 1 + 2

Latent


Exploratory factor analysis (EFA)Exploratory factor analysis (EFA)4

FactorsVar F1 F2 F3 h2

60 06 02 36F1 F2

x1 .60 -.06 .02 .36x2 .81 .12 -.03 .67x3 .17 .73 .08 .60x4 .01 .65 -.04 .424x5 .03 .10 .87 .65x6 .12 .22 .65 .47

x1 x2 x6x3 x4 x5

F3


Confirmatory factor analysis (CFA)Confirmatory factor analysis (CFA)5

FactorsVar F1 F2 F3

60 0 0x1 .60 0 0x2 .81 0 0x3 0 .73 0x4 0 .65 0

F3F1 F2

fit4x5 0 0 .87x6 0 0 .65 x1 x2 x6x3 x4 x5

fit


Confirmatory Factor Analysis (CFA)Confirmatory Factor Analysis (CFA)

x = + Fairness

iTrust in S i

6

x1 = 11 1 + 1

x2 = 21 1 + 2

x3 = 32 2 + 3

perception Supervisor

x3 32 2 + 3

x4 = 42 2 + 4

x1 x2 x3 x4Item2 Item3Item1 Item4

1 x2 x3 x4

1. My supervisor is fair.2. My supervisor treats us without biases.

3. I believe that my supervisor will protect me.4. I trust my supervisor.


Confirmatory Factor Analysis (CFA)y y ( )

x1 = 11 1 + 1

7

x1 11 1 1

x2 = 21 1 + 2

x3 = 32 2 + 31 1 + 3 x4 = 42 2 + 4

x1 x2 x3 x4

Item2 Item3Item1 Item41 x2 x3 x4

No Cross Loadings


No Cross Loadings(cross loading is common in personality inventories)



Confirmatory Factor Analysis (CFA)Confirmatory Factor Analysis (CFA)

x1 = 11 1 + 1

8

1 11 1 1x2 = 21 1 + 2x3 = 32 2 + 3

+

x4 = 42 2 + 4

Let the errors of x2 & x3 correlatex1 x2 x3 x4

Item2 Item3Item1 Item4

1 x2 x3 x4


No correlated errors



Why not correlated errors ?Why not correlated errors ?

+

9

x1 = 11 1 + 1

x2 = 21 1 + 2

x = + x3 = 32 2 + 3

x4 = 42 2 + 4

x1 x2 x3 x4


10

From correlation to Structural Equation Modelingg


Structural ModelS uc u a Mode

Path diagram

11

Path diagram

Job satisfaction Job performancerxy = .36

Predictor variable Criterion variable

xy

Independent variable Dependent variable


Structural Model12

AbilityPersonality

Job satisfaction Job performance

AgegGenderTenure

Criterion variableDependent variable


Structural Model13

AbilityPersonality

Job satisfaction

Job performance

AgeTurnover

gGenderTenure


Path ModelPath Model14

Path Analysis

TransformationalTransformationalLeadership

Leader-memberE h (LMX)

SubordinateExchange (LMX) Performance

TransactionalLeadershipp


Path model vs. Measurement ModelPath model vs. Measurement Model

In a path (structural) model, all the variable are measured, (that is

15

p ( ) (observed and concrete, we assume no measurement errors).

TransformationalLeadership Leader-member Subordinate

In a measurement model, the items are observed and concrete, but the

Exchange (LMX)SubordinatePerformanceTransactional

Leadership

latent variables (constructs) are unobserved and abstract.

Fairness Trust in

perception Supervisor

x1 x2 x3 x4Observed

Latent

x1 x2 x3 x4

Observed


Terminology and symbolsgy yCircle – unobservable constructsRectangle – observable measures

16

Rectangle observable measuresArrow – causal directionDiagram flows from left to rightTwo way curved arrow – unexplained correlations y1

y2

x1

x2

y1 y2

y

Factor loading ( )Latent construct ( )Factor correlation ( ) x3

y3

y4

Measurement error ( , )Estimation error ( )Exogenous variables, (variables which causes are not explained )Endogenous variables (variables which causes are explained in the model )

x4

Endogenous variables, (variables which causes are explained in the model )The effect of exogenous variables on endogenous variables ()The effect of endogenous variables on endogenous variables ( )


Structural Equation ModelingStructural Equation Modeling

Transformational

17

1

1

1x

2x

1y1

1

TransformationalLeadership

2

12y

22

LMXJob performance

3x

23y

4y

3

43

Job performance

2

4x 24

TransactionalLeadershipp



Transformational

18

Leadership

x1

x2

y1

y

LMXJob performance

y2

x3

y3

y4

TransactionalLeadership

x4

y4

Leadership




19

x1Leadership

x2

y1

y

LMXJob performance

y2

Job performance

x3

y3

y4


3

x4

y4

eade s p


Structural Equation ModelingStructural Equation Modeling20


1

1

1x

2x

1y

y

1

1

2

Leadership

1 2

y22

LMXJob performance

3x

2

3y

4y

3

43

2

4x

4

2

4

4



eade s p

21

The relationship between SEM andThe relationship between SEM and other correlational studies


Complexity of analysisComplexity of analysis22

X Y Measurement Relationship TechniqueAmongPredictorsPredictors

1 1 No NoN 1 N N

Correlation

R iN 1 No NoN N No NoN N YES No

Regression

Multivariate Regression

CFA

N N No YESN N YES YES

Path Analysis

SEM


23

How does SEM programs such asHow does SEM programs such as LISREL, EQS, AMOS estimate the parameters?parameters?


A

B = 1 C

Theoretical Model

24

B C

1 2 C = 2

r b = 1 a b c

rAB = .61r = 42

rab 1

rac = 2

rbc = 1 2

a b ca 1 2b 1 12c 2 12

rAC = .42rBC = .35

Ob i Based on the theoretical model, weObservation Based on the theoretical model, we 1. come up with estimates (by simulations) of the

parameters of interest (1 & 2); 2. try to reproduce the observed data using our

ti t

a b ca estimates;

3. if we can perfectly reproduce the observation using our estimates, we have an estimated model with 100% fit. The parameters are assumed to be

b c

phighly believable.


A

B = 1 C

Theoretical Model

25

B C

1 2 C = 2

r b = 1

Model

a b c

rAB = .61r = 42

rab 1

rac = 2

rbc = 1 2

a 1 2b 1 12c 2 12

rAC = .42rBC = .35

Ob i

Problem

1 2 1 2b b Observation

a b ca

bbbb

.61 * .42 = .2562

.65 * .45 = .2925

.63 * .53 = .3339 Optimal (Total

1 2 1 2b b

b c

2 2 21 2 1 2 .61 .42 .35 minimumGoodness of Fit A b b b b

p (error is lowest)

1 1 2 2 1 2 1 2 ; ; b b b b

1 2 1 2 .61 .42 .35 minimumGoodness of Fit B b b b b


A

B = 1 C = 2

26

B C

1 22

rab = 1

Theoretical

rAB = .61r = 42

rac = 2

rbc = 1 2

Theoretical Model

rAC = .42rBC = .35 Observed Est1 Est2 Est3

rAB = .61 Ob i rAC = .42

rBC = .35

G d f fit 0938 0762 0952

Observation

Goodness-of-fit .0938 .0762 .0952

Goodness of Fit = (rab-rab)2 + (rac-rac)2 + (rbc-rbc)2


A = 1

Theoretical Model

27

B C

1 1 C = 2

r b = 1 b

rAB = .61r = 42

rab 1

rac = rbc =

a b ca 1 2b 1 3+12c 2 12 rAC = .42

rBC = .35

Ob i

2 12

ProblemObservationa b c

a

+

Problem.61 * .42 = .2562 ; b3 = .0938b1 * b2 = b1*b2 + b3

Optimal (Total b c

0Goodness of Fit A Goodness of Fit B

p (error is lowest)


Path Model

r < r and r28

X1

r14 < r13 and r34r24 < r23 and r34


Leader-member SubordinateExchange (LMX) Performance

TransactionalX3 X4

LeadershipX2



Transformational

29

1

1

1x

2x

1y

y

11

2

Leadership

1 2

y22

LMX Job performance

3x

2

3y

4y

3

43

2

4x

4

2

4

4

TransactionalLeadershipLeadership


A

30

r b = 1

Theoretical Model a b cB C

1 2

rab 1

rac = 2

rbc = 1 2

a 1 2b 1 12c 2 12 a b c

a a b c

Problem61 * 42 = 2562

Observation

.61 .42 = .2562

.65 * .45 = .2925

.63 * .53 = .3339b1 * b2 = b1*b2

Optimal (Total error is lowest)

2 2 21 2 1 2 .61 .42 .35 minimumGoodness of Fit A b b b b

)

1 2 1 2 .61 .42 .35 minimumGoodness of Fit B b b b b Kenneth Law @ 同济大学 2010

= 1 A

31

1 C = 2

r b = 1 bB C

1

Theoretical Model rab 1

rac = rbc =

a b ca 1 2b 1 3+12c 2 12

a b ca 2 12a b c

Problem

+

Problem.61 * .42 = .2562 ; b3 = .0938b1 * b2 = b1*b2 + b3

Optimal (Total Observation

0Goodness of Fit A Goodness of Fit B

p (error is lowest)


An exampleTrial values Correlations Criterion An example1 rAB rAC rBC d2

Observed .61 .42 .23Iterations1 .5 .5 .50 .50 .25 .018900 (.61-.50)2+(.42-.50)2+(.23-.25)2

32

A

1a .501 .5 .501 .50 .2505 .0187011b .5 .501 .50 .501 .2505 .0190812 .6 .5 .60 .50 .30 .0114002a .601 .5 .601 .50 .3005 .011451

( ) ( ) ( )

B C

1 22b .6 .501 .60 .501 .3006 .0116453 .6 .4 .60 .40 .24 .0006003a .601 .4 .601 .40 .2404 .0005893b .6 .401 .60 .401 .2406 .000573 B C

rAB = .61

4 .6 .41 .60 .41 .246 .0004564a .601 .41 .601 .41 .2464 .0004504b .6 .411 .60 .411 .2466 .0004575 .61 .41 .61 .41 .2501 .000504 AB

rAC = .42rBC = .23

5a .601 .41 .601 .41 .2464 .00045035b .602 .41 .602 .41 .2468 .00044695c .601 .411 .601 .411 .2470 .00045146 .602 .41 .602 .41 .2468 .00044696a .603 .41 .603 .41 .2472 .00044596b .602 .411 .603 .411 .2474 .00044857 .603 .409 .603 .409 .2462 .0004480


An exampleTrial values Correlations Criterion An example

A

1 rAB rAC rBC d2

Observed .61 .42 .23Iterations1 .5 .5 .50 .50 .25 .018900

33

A1 2

1 .5 .5 .50 .50 .25 .0189001a .501 .5 .501 .50 .2505 .0187011b .5 .501 .50 .501 .2505 .0190812 .6 .5 .60 .50 .30 .0114002a .601 .5 .601 .50 .3005 .011451

B C2b .6 .501 .60 .501 .3006 .0116453 .6 .4 .60 .40 .24 .0006003a .601 .4 .601 .40 .2404 .0005893b .6 .401 .60 .401 .2406 .000573

rAB = .61rAC = .42

4 .6 .41 .60 .41 .246 .0004564a .601 .41 .601 .41 .2464 .0004504b .6 .411 .60 .411 .2466 .0004575 .61 .41 .61 .41 .2501 .000504

rBC = .235a .601 .41 .601 .41 .2464 .00045035b .602 .41 .602 .41 .2468 .00044695c .601 .411 .601 .411 .2470 .00045146 .602 .41 .602 .41 .2468 .00044696a .603 .41 .603 .41 .2472 .00044596b .602 .411 .603 .411 .2474 .00044857 .603 .409 .603 .409 .2462 .0004480

Minimum value of the fit function


LISREL Estimation – An ExampleLISREL Estimation An Example34

b b

Observed var-cov matrixb

a c

1 a b ca 1.0b .45 1.0

** * *

ab

bc ab bc

b r ac r b r r a

c .25 .35 1.0

2

var( ) var( )var( ) var( ) var( )ab ab

a ab r a r a

2 2

1 2

var( ) 1var( ) (.45) .20

var( ) 1; .45; .25;

ab

if our estimates are that a thenab r

2 2

2

var( ) var( ) var( )cov( , ) cov( , ) cov( , ) var( )cov( , ) cov( , ) cov( , ) var( )

cov( ) cov( )

ab bc ab bc

ab ab ab

ab bc ab bc ab bc

c r r a r r aa b a r a r a a r aa c a r r a r r a a r r ab c r a r r a r r

2cov( ) var( )a a r r a

2 2

2

2 2

( ) ( )

var( ) var( ) (.45) (.25) .02cov( , ) var( ) .45cov( , ) var( ) (.45)(.25) .16

( ) ( )

ab

ab bc

ab

ab bc

c r r aa b r aa c r r ab

2( 45) ( 25) 07cov( , ) cov( , )ab ab bc ab bb c r a r r a r r cov( , ) var( )c ab bca a r r a 2cov( , ) var( )ab bcb c r r a 2(.45) (.25) .07

var(a) var(b) var(c) cov(ab) cov(ac) cov(bc)

Obs 1.0 1.0 1.0 .45 .25 .35


Est 1.0 .20 .02 .45 .16 .07

.00 .80 .98 .00 .09 .28 Total 2

2 .00 .64 .95 .00 .01 .08 1.67

LISREL Estimation – An ExampleLISREL Estimation An Example35

Ob dvar(a) var(b) var(c) cov(ab) cov(ac) cov(bc)

Observed var-covmatrix

Obs 1.0 1.0 1.0 .45 .25 .35

1 1 .20 .02 .45 .16 .07 1.6735

2 1 1 22 03 50 17 08 1 64222 1.1 .22 .03 .50 .17 .08 1.6422

3 1.2 .24 .03 .54 .19 .09 .16364

4 1.3 .26 .03 .59 .20 .09 1.6564 拟合指数5 1.4 .28 .03 .63 .22 .10 1.7013

6 1.5 .30 .04 .68 .24 .11 1.7719

7 1.6 .32 .04 .72 .25 .11 1.8681

Minimum Estimation Error

b8 1.7 .34 .04 .77 .27 .12 1.9897

9 1.8 .36 .04 .81 .28 .13 2.1367

10 1 9 38 05 86 30 13 2 3093

b

a c

1


10 1.9 .38 .05 .86 .30 .13 2.3093

Estimated var-cov matrix

The covariance matrixThe covariance matrix

Variance covariance matrix ( )

36

X1 X2 X3 X4 Y1 Y2 Y3 Y4X1 1.5X2 .55 1.3X3 .24 .22 2.1

Variance-covariance matrix ( )

1xTransformationalreliability

X4 .19 .21 .60 1.8Y1 .32 .35 .31 .38 1.1Y2 .24 .21 .29 .33 .65 1.4Y3 .11 .10 .38 .42 .44 .42 1.9Y4 .19 .17 .21 .18 .41 .47 .49 1.2

11x

2x

3x

1

2

1y

2y

3yLMX Performance

y

23x

4x4y

Transactional

Actual observations Theoretical relationship


The estimation procedurep37

x1 = 1 1 + 1

x2 = 2 1 + 2

x3 = 3 2 + 3

11x

2x

1

1y

2y

3y3 3 2 3

x4 = 4 2 + 4

y1 = 5 1 + 5

+

23x

4x

2

4y

Estimated structure of

^y2 = 6 1 + 6

y3 = 7 2 + 7

y4 = 8 2 + 8

structure of covariance

matrix

Var(x ) = 2 Var( ) + Var( )1 = 1 1 + 9

2 = 2 1 + 3 2 + 10

Var(x1) = 12 Var(1) + Var(1)

Cov(x1, x2) = Cov([1 1 + 1 ], [3 2 + 3])= 1 3 Cov(1, 2)

10 ……Kenneth Law @ 同济大学 2010

The estimation procedurep

11x

2x1y X1 X2 X3 X4 Y1 Y2 Y3 Y4

38

x1 = 1 1 + 1

x2 = 2 1 + 2

x = + 2

2

3x

1

2

2y

3y

4y

X1 1.5X2 .55 1.3X3 .24 .22 2.1X4 .19 .21 .60 1.8Y1 .32 .35 .31 .38 1.1Y2 .24 .21 .29 .33 .65 1.4x3 = 3 2 + 3

x4 = 4 2 + 4

y1 = 5 1 + 5

4xY2 .24 .21 .29 .33 .65 1.4Y3 .11 .10 .38 .42 .44 .42 1.9Y4 .19 .17 .21 .18 .41 .47 .49 1.2

Var(x1) = 12 Var(1) + Var(1)

Cov(x1, x2) = 1 3 Cov(1, 2)Estimated

structure of iy2 = 6 1 + 6

y3 = 7 2 + 7

y4 = 8 2 + 8

Estimated covariance

matrixObserved covariance

matrix

……covariance matrix

y4 8 2 8

1 = 1 1 + 9

2 = 2 1 + 3 2 + 10 ^

Compare with

matrix matrix

Goodness of Fit Index = | |^Kenneth Law @ 同济大学 2010

A simplified example of fitA simplified example of fit

x = + 11x

1yX1 X2 X3 X4 Y1 Y2 Y3 Y4

X1 1 5

39

x1 = 1 1 + 1

x2 = 2 1 + 2

x3 = 3 2 + 3

1

2

2x

3x

1

2

2y

3y

4y

X1 1.5X2 .24 1.3X3 .13 .22 2.1X4 .19 .21 .60 1.8Y1 .32 .35 .31 .38 1.1Y2 .24 .21 .29 .33 .65 1.4

x4 = 4 2 + 4

y1 = 5 1 + 5

y2 = 6 1 + 6

24x

Observed correlation between and 24

Y3 .11 .10 .38 .42 .44 .42 1.9Y4 .19 .17 .21 .18 .41 .47 .49 1.2

y3 = 7 2 + 7

y4 = 8 2 + 8

1 = 1 1 + 9

x1 and x2 = .24

Estimated correlation between 1 1 1 + 9

2 = 2 1 + 3 2 + 10x1 and x2 based on the estimated set of ( 1, 2, 3, … 1 … 1... ) = .05

G d f Fit I d | |^Goodness of Fit Index = | |


G d Of FiGoodness-Of-Fit

Fit value function (F)

40

Fit value function (F) = (N-1) F

^

Best estimates of parameter set( 1, 2, 3, … 1 … 1... )

( 1, 2, 3, … 1 … 1... )


Goodness of fit indices41

Goodness of fit indices The Goodness of fit index (GFI) do not depend on sample size explicitly (Note:

) d h h b h d l fithe sampling distribution still depends on N) and measure how much better the model fits as compared with no model at all (i.e. all parameters are zero).

ˆ, ( )1

, (0)

F

F

SGFI

S

The Adjusted Goodness of fit index (AGFI) adjusts for degrees of freedom.

( 1)(1 )2

1 k k GFId

AGFI

k is the number of variables; d is the degrees of freedom of the model.

Goodness-of-fit Statistics42

Goodness of fit Statistics

FN

12

( ) 1i

dd

FPNFI Fi

( 1) / /( 1) / 1

i i

i i

N F d F dN F d

NNFI

iFFNFI 1

{( 1) ;0}{( 1) ;0}

1i i

max N F dmax N F d

CFI

i

0ˆ { /( 1);0}F max F d NRMSEA d d

i

{( 1) ;0}i imax N F d d d

• F is the minimum value of the fit function for the estimated model;

• Fi is the minimum value of the fit function for the independence model (when all the correlations and covariances are zero);

• d is the degrees of freedom of the model.g

Transformational

1

1

1x

2x

1y

y

1

1

2


43

1 2

y22

LMX

Job performance

3x

2

3y

4y

3

43


Job performance

2

4x

4

2

4

4

Based on the observed covariance matrix, find a set of best estimates of parameter set ( 1, 2, 3, …… )T i i i th Fit l f ti ^To minimize the Fit value function ^


4444

LISREL PROGRAMLISREL PROGRAM


The SIMPLIS languageg g45


46

Title: Test programVariable:

46

Observed variables:x1 x2 x3 x4 x5 x6Latent Variables: a b

}Latent Variables: a bCorrelation matrix: 1.0.50 1.052 43 1 0 }.52 .43 1.0.22 .16 .15 1.0.12 .31 .20 .69 1.0.26 .13 .29 .66 .72 1.0

Sample size: 139

Raw data from file: test.txt

Your model

pRelationship:x1 x2 x3 = ax4 x5 x6 = bx4 x5 x6 bAdmissibilities = offIterations = 1000P th diPath diagramEnd of Problem


47

Title: Test programVariable: Your model

47

Observed variables:x1 x2 x3 x4 x5 x6Latent Variables: a b

x1 = 1 a + 1x2 = 2 a + 2

+Latent Variables: a bCorrelation matrix: 1.0.50 1.052 43 1 0

x3 = 3 a + 3

x4 = 4 b + 4.52 .43 1.0.22 .16 .15 1.0.12 .31 .20 .69 1.0.26 .13 .29 .66 .72 1.0

Sample size: 139

4 4 4x5 = 5 b + 5x6 = 6 b + 6p

Relationship:x1 x2 x3 = ax4 x5 x6 = b} a bx4 x5 x6 bAdmissibilities = offIterations = 1000P th di

}x x xPath diagram

End of Problemx1 x2 x3 x4 x5 x6


4848

LISREL OutputLISREL Output


49

Your model

49

Your model

x1 = 1 a + 1x2 = 2 a + 2x2 2 a + 2x3 = 3 a + 3

b +x4 = 4 b + 4x5 = 5 b + 5x6 = 6 b + 6

a b

x1 x2 x3 x4 x5 x6


5050


5151


52

Your model

52

x1 = 1 a + 1x2 = 2 a + 2

+x3 = 3 a + 3

x4 = 4 b + 44 4 4x5 = 5 b + 5x6 = 6 b + 6

a b

x x x

.76

x1 x2 x3 x4 x5 x6


5353

Model 2

RMSEA(Root Mean Square Error of

i i )Approximation)

Tucker-Lewis Index (TLI, NNFI)

Incremental Fit Index (CFI)


5454

a b

x1 x2 x3 x4 x5 x6

The Modification Indices Suggest to Add an Error Covariance

Between and Decrease in Chi-Square New Estimate

x5 x1 15.4 -0.17

x5 x2 20 6 0 21x5 x2 20.6 0.21

x6 x2 13.1 -0.17


Scaling in SEMScaling in SEM55

x1 x2 x3 x4 x1 x2 x3 x4


Scaling in SEMScaling in SEM56

x1 = 11 1 + 1

+

x2 = 21 1 + 2

x3 = 32 2 + 3

x = +

x4 = 42 2 + 4x1 x2 x3 x4


Scale by the first item of each factorScale by the first item of each factor57

x1 = 11 1 + 1

+

x2 = 21 1 + 2

x3 = 32 2 + 3

x = +

x4 = 42 2 + 4

x1 x2 x3 x4


Scale by standardization58

Scale by standardization

x1 = 11 1 + 1

x = + 2

2 1 1

2 1

x2 = 21 1 + 2

x3 = 32 2 + 3

x = +

x4 = 42 2 + 4x1 x2 x3 x4


LISREL Programming – ScalingLISREL Programming Scaling59

Relationships:

obsx1 obsx2 = latent X

1*obsx1 obsx2 = latent X1 obsx1 obsx2 latent X

(.5) obsx1 obsx2 = latent X


Other issues in SEM6060

Other issues in SEM

1. Degrees of freedom2. Identification3 Nested model3. Nested model4. Missing data in SEM5. Single indicatorg6. Parceling of indicators7. Factorial Invariance in cross-cultural research



Other issues in SEM



Degrees of freedomg

62

x1 x2 x3

x4 x5 x6

x7 x8 x9

Number of variance-covariance terms =Number of parameters to be estimated =D f f d

9(10)/2 = 4521

Degrees of freedom = 45-21 = 24


Degrees of freedomg

63

x1 x2 x3

N b f i i t 3(4)/2 6Number of variance-covariance terms =Number of parameters to be estimated =Degrees of freedom =

3(4)/2 = 67

– 1



Other issues in SEM



The issue of Identification in SEM65

Fairness Trust in x Fairnessperception Supervisor

x1 x2 x3 x4

x

x x x x

x1 1 .3 .4 .6x2 .3 1 .5 .7x3 .4 .5 1 .3x1 x2 x3 x4

x3 .4 .5 1 .3x4 .6 .7 .3 1


10 observed statistics9 estimated parameters



Identification in SEM66

Fairnessperception

Trust in Supervisor

x1 x2 x3 x4 x5x1 1 .3 .4 .6 .3

3 1 5 7 4

x2 .3 1 .5 .7 .4x3 .4 .5 1 .3 .6x4 .6 .7 .3 1 .5

x1 x2 x3 x4

4x5 .3 .4 .6 .5 1

Observed statistics = 10 Observed statisticsEstimated parameters =

1011


Identification in SEM

F i Trust in

67

Fairnessperception Supervisor

x1 x2 x3 x4 x5x1 1 .3 .4 .6 .3x2 .3 1 .5 .7 .4

x

2x3 .4 .5 1 .3 .6x4 .6 .7 .3 1 .5x 3 4 6 5 1

x1 x2 x3 x4

x5 .3 .4 .6 .5 1

15 observed statistics13 ti t d t

x5

13 estimated parameters



Other issues in SEM



6969

Testing nested modelTesting nested model


Testing nested modelsg70

Two models are nested within each other if they are exactly the same except that some parameters in one model are fixedparameters in one model are fixed.

The model with parameters fixed is said to be t d ithi th d l ith t tnested within the model without parameters

fixed.


71

Transformational

71


LMXSubordinatePerformance

Transactional

Model 1




Transactional

Model 2

Leadership

Transformational

Model 2 is nested within model 1



Transactional

Model 2

Leadership



x x x x xx x x x x x1 x2 x3 x4 x5

x1 x2 x3 x4

x5x1 x2 x3 x4

x5

x1 x2 x3 x4

x5

M d l 2 i t d ithi d l 1

Model 2Model 1 Model 3


Model 2 is nested within model 1

Model 3 is not nested within model 1

Are these two models nested?73

x x x x x x x xx1 x2 x3 x4

x1 x2 x3 x4

A one-factor model versus a two-factor model


Are these two models nested?74

x1 x2 x3 x4

x1 x2 x3 x4


7575

The differences in model of two nested models is also a distribution, with degrees of freedom equals the difference in d f of the two nested modelsd.f. of the two nested models.

Using this difference in distribution, we can then test whether fixing these parameters in the nested model will cause statistically significant change in model fit.

If model fit () changes significantly after fixing these parameters, it implies that the parameters should not be fixed.Conversely if model fit does not change significantly by fixingConversely, if model fit does not change significantly by fixing these parameters, it would be acceptable to fix them.


76

Transformational

76



Transactional

d.f.


Transformational TransformationalLeadership


Transactional

d.f.

Leadership



Transactional

Leadership



x7 x8 x9 x10x7 x8 x9 x10

ax1

x2

c

x3 b

x4 x5 x6


78

x8 x10x7 x9

78

Model 1: 1

2 df1ac

x2

x1

bx3

2 = 12 - 2

2

d.f. = df1 - df2

x5x4 x6

x8 x10x7 x9

cx1

x8 x10x7 x9

a

b Model 2: 2

2 df2

x2

x3

x5x4 x6



Model 1:Relationships:x1 x2 x3 = ax4 x5 x6 = b

x8 x10x7 x9

a

bcx2

x1

x3x4 x5 x6 = bx7 x8 x9 x10 = cb = ac = b

x5x4 x6

b3

2 = 12 - 2

2

Model 2:Relationships:x1 x2 x3 = a x8 x10x7 x9

1 2

d.f. = df1 - df2

x4 x5 x6 = bx7 x8 x9 x10 = cb = ac = a b x5x4 x6

a

bcx2

x1

x3


c = a b 54 6


Other issues in SEM



Missing data in SEMg81

1 2 3 4 5 6x1 x2 x3 x4 x5 x6

1 3 4 3 2 4 62 5 4 3 4 3 73 3 2 1 5 4 34 2 4 3 -- 6 55 4 3 2 1 5 36 6 4 3 2 1 26 6 4 3 2 1 27 6 5 4 3 1 48 3 5 2 1 5 39 4 5 1 3 2 610 5 4 1 2 3 410 5 4 1 2 3 4

Casewise (listwise) deletion of missing data


Pairwise deletion of missing data

Missing data in SEMg82

t t b i + b lt+ b + f + fb+ f +ti +tib+ticompute temp=ocbciv+ocbalt+ocbcon+perfa+perfb+perfc+tia+tib+tic.

**The above statement will assign a missing value to the variable temp if anyone of the 12 variables is missing.

**An alternative way is to define temp=(ocbciv+cobalt+ … +tic)/9. Temp is undefined with i i i bl d h f i i l ill b i d

y p ( ) pany missing variable and, therefore, a system missing value will be assigned to temp whenever anyone of the 9 variable is missing.

Missing value temp (99).select if (temp ne 99).( p )

**Select if temp<99 means that we will exclude all cases when temp has a missing value. We can then write the data to a text file to be used as the LISREL data file.

it tfil 'C \ 24 t t'write outfile='C:\sep24.txt'/ocbciv ocbalt ocbcon perf1 perf2 perf3 ti1 ti2 ti3 (9(F4.2,1X)).

execute.



Other issues in SEM



Single indicators in LISRELg84

x8 x10x7 x9

x7


Job Satisfaction

x2

x1

LMXx3

x5x4 x6


Single indicators in LISRELg85

ttCov )( 2

x t Job Satisfaction

t

xtxtxx

ttCov

ttCovrr

2),(

),(

22

2

x

x

txxr

x7

xxt

xx

xt

t

xtxx

rr

r

)(

1 2

2

2x

xxx

r

r

x7

xx

xxx

xx

r

)1(

1

22

2

xxx

xxx

r

r

)1(22 x


86

Title: Test program

86

Variable: Observed variables:x1 x2 x3 x4 x5 x6 x7L V i bl bLatent Variables: a b cCorrelation matrix: 1.0.50 1.052 43 1 0.52 .43 1.0.22 .16 .15 1.0.12 .31 .20 .69 1.0.26 .13 .29 .66 .72 1.0.24 .11 .25 .31 .27 .38 1.0

Sample size: 139Sample size: 139Relationship:x1 x2 x3 = ax4 x5 x6 = b

Single indicator

}x4 x5 x6 bx7 = cLet the path c -> x7 be .9220Let the error variance of x7 be .1278}End of Problem



Other issues in SEM



Parceling of indicatorsg88

• M l i OB/HR h it (i di t )• Many scales in OB/HR have many items (indicators)• For example, a simple model of justice perceptions

predicting OCB may involve 6 procedural justice items, p g y p j ,6 interactive justice items, 5 distributive justice items note1

and 20 items measuring OCB. note 2

• Th t t l b f i di t 37 lti i 703• The total number of indicators are 37, resulting in 703variance-covariance terms.

Note 1: Moorman, R.H. (1991) Relationship between organizational justice and organizational citizenship behaviors: do fairness perceptions influence employee citizenship? Journal of Applied Psychology, 76(6), 845-855.

Note 2: Farh, J., Earley, P.C., Lin, S. (1997). Impetus for action: A cultural analysis of justice and organizational citizenship behavior in Chinese society. Administrative Science Quarterly, 42(3), 421-444.


References89

Mathieu, J. E. & Farr, J. L. (1991). Further evidence for the discriminant validity of measures of organizational y gcommitment, job involvement, and job satisfaction. Journal of Applied Psychology, 76, 127-133.

Mathieu, J. E., Hofmann, D. A., & Farr, J. L. (1993). Job perception-job satisfaction relations: An empirical comparison of three competing theories. Organizational Behavior and Human Decision Processes, 56, 370-387.


The research questionsq90

1 T i ti t h th j b ti f ti j b1. To investigate whether job satisfaction, job involvement and organizational commitment are distinct constructs

2. This paper:a) Shows the idea of discriminant validitya) Shows the idea of discriminant validityb) Illustrate the use of CFAc) Illustrate how reduce indicators can be

achieved when sample size is smallachieved when sample size is small

Mathieu, J. E. & Farr, J. L. (1991). Further evidence for the discriminant validity of measures of organizational i j b i l d j b i f i l f A li d h l 6 12 133


commitment, job involvement, and job satisfaction. Journal of Applied Psychology, 76, 127-133.

Study Oney91

• N1=194 bus driversO i i l C i d b 9 i h f f P S• Organizational Commitment measured by 9-item short form of Porter, Steers, Mowday & Boulian’s (1974) scale

• Job involvement measured by 6 items selected from Lodahl and Kejner’s (1965) scaleJ b i f i d b h 20 i i f h MSQ• Job satisfaction measured by the 20-item version of the MSQ

• Role strain assessed by 12-item selected from House, Schuler, & Levanoni (1983)• Role conflict by 6-item, role ambiguity measured by 6-items, the two aggregated to

form role conflict/ambiguity. J b i t k id tit kill i t t f db k d i t ti• Job scope measuring task identity, skill variety, autonomy, feedback and interaction facilitation by 15-items drawn from Stone (1974) and Sims, Szilagyi, & Keller (1976)

• “Garage pride” (6 items from Jones & James, 1979) measures the extent to which d i d f th i hi h th kdrivers are proud of the garage in which they work;

• Job tension represents the extent to which factors related to drivers’ jobs afect their health and well-being (7-item drawn from House & Rizzo, 1972)

• Human Resources Management represents drivers’ perception of the management of th (14 it d t d f T l & B 1972)


the company (14 items adapted from Taylor & Bowers, 1972)

Study Twoy92

• N2=483 engineers• Organizational Commitment measured by 15-item short form of Porter, Steers,

Mowday & Boulian’s (1974) scale• Job involvement measured by 6 items selected from Lodahl and Kejner’s (1965)

scale• Job satisfaction measured by 15 items by Hackman & Oldham (1974)• Job scope measured by 35 items from Sims et al. (1976) and Withey, Daft & Cooper

(1983)• Self- and supervisor performance ratings by 13-Behavior Anchored Rating Scale

items• Education level, position tenure, organizational tenure and age as control


The procedurep93

Three indicators were established for eachOC01 .45OC02 .23 4thThree indicators were established for each

multi-item measure by first fitting a single factor solution to each set of items and then averaging the items with highest and

OC02 .23OC03 .28OC04 .81OC05 89 1st

4th

lowest loadings to form the first indicators, averaging the items with the next highest and lowest loadings to form the second i di t d f th til ll it

OC05 .89OC06 .76OC07 .55

1st

2 dindicator and so forth until all items were assigned to one of the three indicators for each variable. This procedure was necessary to reduce the number of

OC08 .14OC09 .29OC10 .86

2nd

3rdnecessary to reduce the number of parameters estimated in the measurement models.

OC11 .75OC12 .84OC13 .34

5th

OC14 .66OC15 .27

NewOC1=(OC02+OC05+OC08+OC10+OC12)/5

Analysisy94

Two sets of analyses were performed. y p

The first set of analyses involved a comparison of the relative fit of three-, two-, and single-factor measurement models. The three-factor model l d h h i di f OC d S l f hplaced the three indicators of OC, JI and JS on separate latent factors. The

three two-factor models were established by forcing the three indicators of two constructs to a single factor and placing the three indicators from the remaining construct on a second factor. The single-factor model forced all g gnine indicators onto a single latent factor.

The second set of analyses examined the relationship between a set of correlates and commitment, job satisfaction, and job involvement. This was accomplished by comparing the relative fit between two models: one in which the relationships among the three variables were freely estimated, and one in which the relationships between each correlate and the three


pvariables were specified as being equal.

Analysesy95

OC JI JS

r1

Criterion variables

Criterion variables

rxyr1

r1


Analyses Bus Driver Sample p.128 Ri ht H d C lRight-Hand-Column

96

(9*10)/2=45 var-covar terms3

99

99 99d.f. = 45 – 21 = 24 d.f. = 45 – 18 = 27


d.f. = 27 – 24 = 3; 2(3,N=194)=196.44***

Analyses Bus Driver Sample p.129 LHSy p p97

(28*29)/2=406 var-covar terms

OC JI JSOC JI JSRole strainJob scope

dGarage prideJob tensionHRMSexChildrenMarital status


Marital statusSeniority

Analyses Bus Driver Sample p.129 LHSy p p98

(28*29)/2=406 var-covar terms

OC JI JS

9

915Why not 19?

( )d.f.=406-21-30-36-27=292

OC JI JSRole strainJob scope

d

9y=3

Why not 19?

Garage prideJob tensionHRM 15 xy=3*9=27SexChildrenMarital status

15

= C =36

xy


Marital statusSeniority

x=9C2=36

Analyses Bus Driver Sample p.129 LHCy p p99

(28*29)/2=406 var-covar terms(28 29)/2 406 var-covar termsd.f.=406-21-30-36-27=292

d f t i d d l 292 + 18 310d.f. constrained model = 292 + 18 = 310;

d.f. = 2(18,N=194)=76.20**



Other issues in SEM



Factorial Invariance101

1 2

1 2

USA PRC

• Same structure• Same

S &


• Same & • Same for everything

SIMPLIS Programg102

Group: Academic reading and writing, Grades 5Observed Variables: read5 write5Covariance Matrix:281.349184.219 182.821Means262.236 258.788262.236 258.788Sample Size: 373Latent Variable: verbal5Relationships:read5 = const + 1*verbal5

it 5 t + (1)* b l5write5 = const + (1)*verbal5

Group: Non-academic reading and writing, Grades 5 and 7 Covariance Matrix:174.485134.468 161.869Means248.675 246.896Sample size: 249Relationships:


Relationships:Set the Error Variance of read5 – write5 freeSet the Variance of verbal5 freeEnd of Problem

103

Illustrative exercises in SEMIllustrative exercises in SEM


Confirmatory Factor Analysisy y104

• Data file: cfa.txt• No. of variables = 22• As ordered in the data file:

○ guanxi 1 to 6○ lmx 1 to 7

GX LMXCom Perf

○ lmx 1 to 7○ commitment 1 to 5○ performance 1 to 4

d l

Inter-factor correlations ()guanxi lmx com perf

1.0

Model = 566.37d.f. = 203RMSEA = .11NFI = .92

.70 1.0

.69 .74 1.0

.73 .65 .82 1.0

N .9NNFI = .94CFI = .95SRMR = .076


Path Analysisy105

LMXGX• Data file: cfa.txt• Reduce all the variables to single indicator• use the four single-indicator constructs to

th th d l th i ht

LMX

PerfCom

GX

run the path model on the right

d l Model = 12.35d.f. = 1RMSEA = .27NFI = .97

LMXGX.34**

.32**=.77n.s.

N .97NNFI = .83CFI = .97SRMR = .031

PerfCom

.32

.85**


Structural Equation Modelingq g106

Correlation and Regression Covariance Structure Analysis, Structural

Equation Modeling and LISRELEquation Modeling and LISREL What is SEM? How are estimations obtained? The idea of goodness-of-fit Model Identification Testing nested models Testing nested models


107

Th dThe End


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