Confirmatory Factor Analysis

6.1 6.2 6.3 () 6.4

6.1 CFA(SEM with latent variables)CFASEM

CFA

CFA

6.1.1 ()

Benjamin and Podolny(1996)10731()~7()(59)5

Table 6.1 (59)CFA()CFA() X1 X2 X3 X4 X5 X1 1.00000 0.76490 0.67821 0.67515 0.68186 X2 0.76490 1.00000 0.73522 0.62564 0.76585 X3 0.67821 0.73522 1.00000 0.63170 0.71356 X4 0.67515 0.62564 0.63170 1.00000 0.51748 X5 0.68186 0.76585 0.71356 0.51748 1.00000

()(construct) CFA(2)(GFIAGFI)(reliability)(validity)

(>0.7) (convergent validity)(construct, concept, or research variables)(Likert scale, Stapel scale, or semantic differential)(discriminant validity)(construct, concept)

Multitraitmultimethod matrix () 6.1 1. Strongly Generally Moderately Moderately Generally Strongly Agree Agree Agree Disagree Disagree Disagree Selection is wide. ____ ____ ____ ____ ____ ____ 2. Extremely Quite Slight Slight Quite ExremelyWide Selection ____ ____ ____ ____ ____ ____ Limited Selection 3. +3 ___ +2 ___ +1 ___ Wide Selection -1 ___ -2 ___ -3 ___

6.2 ((store appearance, A and product assortment, P)(Likert scale, L, Differential scale, D, and Stapel scale, S) )

AL AD AS PL PD PS AL 1.000 AD 0.776 1.000 AS 0.676 0.739 1.000 PL 0.638 0.600 0.539 1.000 PD 0.561 0.635 0.527 0.713 1.000 PS 0.522 0.559 0.589 0.720 0.698 1.000

6.2.1 Intuition

1.000 .722 .714 .203 .095 .722 1.000 .685 .246 .181R = .714 .685 1.000 .170 .113 .203 .246 .170 1.000 .585 .095 .181 .113 .585 1.000

X1 = 111 + 122 + 1X2 = 211 + 222 + 2X3 = 311 + 322 + 3 (6.1)X4 = 411 + 422 + 4X5 = 511 + 522 + 5X1 = 111 + 1X2 = 211 + 2X3 = 311 + 3 (6.2)X4 = +422 + 4X5 = +522 + 5 Corr(1,2) = 12, var(1)=11, var(2)=22

CFA(scaling indeterminancy)Var(i)ij:1, 1

11 0 21 0factor loadings matrix = 31 0 0 42 0 52

factor correlation matrix 1 12 21 1

(p.181)

(p.183)

(6.5)

(6.4)~

the proposed model fits as well as a perfect model

Measure Reliability ():

Test-retestCFA 2>0.7

6.2.2 Mechanism (P.184) (6.6)

(6.7)

Likelihood function Xi~N(0, )

(6.10)

(6.11)

(6.12)Final version of the log likelihood functionObtain parameter estimates to maximize 6.12MLE

H0: Reduced model is indifferent from full modelHa: two models are significantly differentSet =0.2

n

(6.16)(6.17)>0.95 good fit>0.9 acceptable fit>0.9 good fit>0.8 acceptable fit

Sample Problems

LisrelLISRELSEMLISREL 8.7

: (P.181)

Title Confirmatory Factor Analysis for student test performanceObserved Variables Correlation Matrix= 1 0.722 1 0.714 0.685 1 0.203 0.246 0.170 1 0.095 0.181 0.113 0.585 1Sample Size=145Latent Variables Relationships: = = = = =SET the Covariance of and to 1 Path DiagramLISREL OUTPUT SE TV RS MI

SE: TV: tRS: QMI: : (P.181)

: (P.181)

GFI=0.99AGFI=0.97RMR=0.022=2.93: (P.181)

Questions Regarding the Application of CFA

Cronbachs alpha (6.18) 0~1item

(6.19)6.4.1 Average inter-item correlation among k itemsitem

: (P.172)Benjamin and Podolny(1996)10731()~7()(59)5

595status

(P.188)2 GFIAGFIRMRSingle dimension

12

3

45Cronbachs =0.91

:(P.188)Title Confirmatory Factor Analysis for Wine IndustryObserved Variables expert1 expert2 expert3 expert4 expert5Correlation Matrix= 1 0.765 1 0.678 0.735 1 0.675 0.626 0.632 1 0.682 0.766 0.714 0.517 1Sample Size=59Latent Variables StatusRelationships: expert1=Status expert2=Status expert3=Status expert4=Status expert5=StatusPath DiagramLISREL OUTPUT SE TV RS MITitle:

:

SE: TV: tRS: QMI:

:(P.189)

:(p.189)t:t>2t:t>2

: (p.189)GFI = 0.96 >0.95 ()AGFI= 0.87 >0.8 ()RMR = 0.031

6.4.2 CFA(P.190) (P.191)

: (P.191) (P.180) 145 , , , , ,

(P.192)2 GFIAGFI=1Single dimension

12

3

45

Title Confirmatory Factor Analysis for student test performanceObserved Variables Correlation Matrix= 1 0.722 1 0.714 0.685 1 0.203 0.246 0.170 1 0.095 0.181 0.113 0.585 1Sample Size=145Latent Variables Relationships: = = = = =SET the Covariance of and to 1 Path DiagramLISREL OUTPUT SE TV RS MI

: (P.192)1 SE: TV: tRS: QMI:

: (P.192)

: (P.192)t::0.07t>2:0.09t

: (P.192)2 = 59.47GFI = 0.88

6.4.2 (P.192)

: (P.193)(P.189)

595status

(P.193)statusexpert3expert2expert1expert5expert42 GFIAGFIRMR12

3

45

: (P.193)Title Confirmatory Factor Analysis for Wine Inc. - restricted model Observed Variables expert1 expert2 expert3 expert4 expert5Correlation Matrix= 1 0.765 1 0.678 0.735 1 0.675 0.626 0.632 1 0.682 0.766 0.714 0.517 1Sample Size=59Latent Variables StatusRelationships: expert1=Status expert2=Status expert3=Status expert4=Status expert5=StatusSET the path from Status to expert1 equal to the path from Status to expert2SET the path from Status to expert2 equal to the path from Status to expert3SET the path from Status to expert3 equal to the path from Status to expert4SET the path from Status to expert4 equal to the path from Status to expert5Equal error variances: expert1 expert2 expert3 expert4 expert5Path DiagramLISREL OUTPUT SE TV RS MI

SE: TV: tRS: MI:

: (P.193)

: (P.193)tt

: (P.193)GFI (acceptable)AGFI (acceptable)RMR

(p.193)

6.4.3 (P.193)

: (p.194) Menezes and Elber(1979)(Likert Scale)(Semantic differential Scale)(Stapel Scale)(store appearance)(product assortment)250

250 ALADASPLPDPS store appearanceproduct assortment

-Simple Model (P.194)appearanceAL2 GFIAGFIassortmentADASPLPDPS12

3

456

Title Confirmatory Factor Analysis for Store Grocery- Simple modelObserved Variables AL AD AS PL PD PSCorrelation Matrix= 1.000 0.776 1.000 0.676 0.739 1.000 0.638 0.600 0.539 1.000 0.561 0.635 0.527 0.713 1.000 0.522 0.559 0.589 0.720 0.698 1.000Sample Size=250Latent Variables APPEARANCE PRODUCTRelationships: AL=APPEARANCE AD=APPEARANCE AS=APPEARANCE PL=PRODUCT PD=PRODUCT PS=PRODUCTPath DiagramLISREL OUTPUT SE TV RS MI

SE: TV: tRS: QMI: :-Simple Model

:- Simple Model (P.195)

t:-Simple Model (P.195)

:- Simple Model (P.195) tAsymptotically Standardized Residual MatrixGFI (good)AGFI (acceptable)

:-Method Factors (P.194) (multitrait, multimethed model,MTMM)

(convergent validity) (discriminant)()

-Method Factors(P.197)appearanceAL2 GFIAGFIproductADASPLPDPS(p.194)

:-Method Factors (P.196)

:-Method Factors (P.195)

:-Method Factors (P.195)GFIAGFIRMR

: -Correlated Error (P.197) Simple ModelP.197

- Correlated Error (P.198)appearanceALproductADASPLPDPS12

3

456

:- Correlated ErrorTitle EX1: Confirmatory Factor Analysis for grocery store-correlated errorObserved Variables AL AD AS PL PD PS Correlation Matrix= 1.000 0.776 1.000 0.676 0.739 1.000 0.638 0.600 0.539 1.000 0.561 0.635 0.527 0.713 1.000 0.522 0.559 0.589 0.720 0.698 1.000Sample Size=250Latent Variables APPEARANCE PRODUCTRelationships: AL=APPEARANCE AD=APPEARANCE AS=APPEARANCE PL=PRODUCT PD=PRODUCT PS=PRODUCTSET the errors between AL and PL correlateSET the errors between AD and PD correlateSET the errors between AS and PS correlate Path DiagramLISREL OUTPUT SE TV RS MI

:- Correlated Error (P.195)

:- Correlated Error (P.195)t

:- Correlated Error (P.195)tt

:- Correlated Error (P.195)GFI(good)AGFI(good)RMR

~ The End ~

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Confirmatory Factor Analysis