confirmatory factor analysis

Confirmatory factor analysis

Hans BaumgartnerPenn State University


x1 x2 x3 x4 x5 x6 x7 x8

What’s the structure underlying 28 distinct covariances between

8 observed variables?


The exploratory factor model

d1 d2 d3 d4 d5 d6 d7 d8

x1 x2 x3 x4 x5 x6 x7 x8

x1 xq

q11d q22

d q33d q44

d q55d q66

d q77d q88

d

11

. . . . . . . . . . . . . . .


The confirmatory factor model

d1 d2 d3 d4 d5 d6 d7 d8

x1 x2 x3 x4 x5 x6 x7 x8

l11 l41l21 l31 l52 l82l62 l72

x1 x2

j21

q11d q22

d q33d q44

d q55d q66

d q77d q88

d

11


d1 d2 d3 d4 d5 d6 d7 d8

x1 x2 x3 x4 x5 x6 x7 x8

l11 l41l21 l31 l52 l82l62 l72

x1 x2

j21

q11d q22

d q33d q44

d q55d q66

d q77d q88

d

11


d1 d2 d3 d4 d5 d6 d7 d8

x1 x2 x3 x4 x5 x6 x7 x8

l11 l41l21 l31 l52 l82l62 l72

x1

j21

q11d q22

d q33d q44

d q55d q66

d q77d q88

d

11

x2


d1 d2 d3 d4 d5 d6 d7 d8

x1 x2 x3 x4 x5 x6 x7 x8

l11 l41l21 l31 l52 l82l62 l72

x1 x2

j21

q11d q22

d q33d q44

d q55d q66

d q77d q88

d

11


d1 d2 d3 d4 d5 d6 d7 d8

x1 x2 x3 x4 x5 x6 x7 x8

l11 l41l21 l31 l52 l82l62 l72

x1 x2

j21

q11d q22

d q33d q44

d q55d q66

d q77d q88

d

11

The congeneric factor model


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21 8811 .... Diag

Appendix A:


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72 11 21 72 21 21 72 31 21 72 41 21 72 52 72 62 72

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82 11 21 82 21 21 82 31 21 82 41 21 82 52 82 62 82 72 82

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21 22

31 32 33

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51 52 53 54 55

61 62 63 64 65 66

71 72 73 74 75 76 77

81 82 83 84 85 86 87 88


d1 d2 d3 d4 d5 d6 d7 d8

x1 x2 x3 x4 x5 x6 x7 x8

l1 l1l1 l1 l52 l82l62 l72

x1 x2

j21

q11d q22

d q33d q44

d q55d q66

d q77d q88

d

11


d1 d2 d3 d4 d5 d6 d7 d8

x1 x2 x3 x4 x5 x6 x7 x8

l11 l41l21 l31 l52 l82l62 l72

x1 x2

j21

q11d q22

d q33d q44

d q55d q66

d q77d q88

d

11

l51


d1 d2 d3 d4 d5 d6 d7 d8

x1 x2 x3 x4 x5 x6 x7 x8

l11 l41l21 l31 l52 l82l62 l72

x1 x2

21= 0 or 1

q11d q22

d q33d q44

d q55d q66

d q77d q88

d

11


d1 d1 d1 d1 d5 d6 d7 d8

x1 x2 x3 x4 x5 x6 x7 x8

l1 l1l1 l1 l52 l82l62 l72

x1 x2

j21

q11 q11 q11 q11 q55d q66

d q77d q88

d

11


d1 d2 d3 d4 d5 d6 d7 d8

x1 x2 x3 x4 x5 x6 x7 x8

l11 l41l21 l31 l52 l82l62 l72

x1 x2

j21

q11d q22

d q33d q44

d q55d q66

d q77d q88

d

11

q51


T1M1 T1M2 T1M3 T2M1 T2M2 T2M3 T3M1 T3M2 T3M3

T1 T2T3

M1 M2 M3

A MTMM model (Appendix B)


Model identification

Setting the scale of the latent variables A necessary condition for identification Two- and three-indicator rules Identification from first principles Empirical identification tests


d1 d2 d3 d4

x1 x2 x3 x4

l11 l21 l32 l42

x1 x2

j21

Identification of a simple model(Appendix C)


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442423242212142211142

33232212132211132

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44332211 Diag

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Identification of a simple model


Model estimation Discrepancy functions:

□ Unweighted least squares (ULS)□ Maximum likelihood (ML)□ Generalized least squares (GLS)□ Other methods

Estimation problems:□ Nonconvergence□ Improper solutions


Model testing Global fit measures:

□ χ2 goodness of fit test□ Alternative fit indices

Local fit measures:□ Parameter estimates□ Reliability□ Discriminant validity

Model modification:□ Modification indices and EPC’s□ Residuals


Reliability for congeneric measures

· individual-item reliability (squared correlation between a construct ξj and one of its indicators xi):

ρii = λij2var(ξj)/[ λij

2 var(ξj) + θii]

composite reliability (squared correlation between a construct and an unweighted composite of its indicators x = x1 + x2 + ... + xK):

ρc = (Σλij)2 var(ξj)/[ (Σλij)2 var(ξj) + Σθii]

average variance extracted (proportion of the total variance in all indicators of a construct accounted for by the construct; see Fornell and Larcker 1981):

ρave = (Σλij2) var(ξj)/[ (Σλij

2) var(ξj) + Σθii]


SIMPLIS specification

Title Confirmatory factor model (attitude toward using coupons

measured at two points in time) Observed Variables id aa1t1 aa2t1 aa3t1 aa4t1 aa1t2 aa2t2 aa3t2 aa4t2 Raw Data from File=d:\m554\eden\cfa.dat Latent Variables AAT1 AAT2 Sample Size 250 Relationships aa1t1 aa2t1 aa3t1 aa4t1 = AAT1 aa1t2 aa2t2 aa3t2 aa4t2 = AAT2 Set the Variance of AAT1 AAT2 to 1 Options sc rs mi wp End of Problem

Confirmatory factor analysis Goodness of Fit Statistics: Degrees of Freedom = 19 Minimum Fit Function Chi-Square = 72.98 (P = 0.00) Normal Theory Weighted Least Squares Chi-Square = 67.97 (P = 0.00) Estimated Non-centrality Parameter (NCP) = 48.97 90 Percent Confidence Interval for NCP = (27.51 ; 78.01) Minimum Fit Function Value = 0.29 Population Discrepancy Function Value (F0) = 0.20 90 Percent Confidence Interval for F0 = (0.11 ; 0.31) Root Mean Square Error of Approximation (RMSEA) = 0.10 90 Percent Confidence Interval for RMSEA = (0.076 ; 0.13) P-Value for Test of Close Fit (RMSEA < 0.05) = 0.00073 Expected Cross-Validation Index (ECVI) = 0.41 90 Percent Confidence Interval for ECVI = (0.32 ; 0.53) ECVI for Saturated Model = 0.29 ECVI for Independence Model = 12.05 Chi-Square for Independence Model with 28 Degrees of Freedom = 2983.93 Independence AIC = 2999.93 Model AIC = 101.97 Saturated AIC = 72.00 Independence CAIC = 3036.10 Model CAIC = 178.83 Saturated CAIC = 234.77 Normed Fit Index (NFI) = 0.98 Non-Normed Fit Index (NNFI) = 0.97 Parsimony Normed Fit Index (PNFI) = 0.66 Comparative Fit Index (CFI) = 0.98 Incremental Fit Index (IFI) = 0.98 Relative Fit Index (RFI) = 0.96 Critical N (CN) = 124.47 Root Mean Square Residual (RMR) = 0.055 Standardized RMR = 0.032 Goodness of Fit Index (GFI) = 0.94 Adjusted Goodness of Fit Index (AGFI) = 0.88 Parsimony Goodness of Fit Index (PGFI) = 0.49

Confirmatory factor analysisMeasurement Equations:

aa1t1 = 1.10*AAT1, Errorvar.= 0.65 , R2 = 0.65 (0.073) (0.070) 15.02 9.23 aa2t1 = 1.10*AAT1, Errorvar.= 0.50 , R2 = 0.71 (0.069) (0.058) 16.06 8.60 aa3t1 = 0.94*AAT1, Errorvar.= 0.74 , R2 = 0.54 (0.071) (0.074) 13.18 9.94 aa4t1 = 1.21*AAT1, Errorvar.= 0.56 , R2 = 0.73 (0.074) (0.066) 16.33 8.40 aa1t2 = 1.20*AAT2, Errorvar.= 0.55 , R2 = 0.72 (0.073) (0.061) 16.43 8.92 aa2t2 = 1.16*AAT2, Errorvar.= 0.41 , R2 = 0.77 (0.068) (0.049) 17.23 8.33 aa3t2 = 0.99*AAT2, Errorvar.= 0.55 , R2 = 0.64 (0.066) (0.056) 14.95 9.65 aa4t2 = 1.23*AAT2, Errorvar.= 0.49 , R2 = 0.76 (0.072) (0.057) 17.04 8.49


Correlation Matrix of Independent Variables

(AVE)½ [.81] [.85] AAT1 AAT2 -------- --------[.81] AAT1 1.00 [.85] AAT2 0.90 1.00 (0.02) 45.14


Standardized Residuals aa1t1 aa2t1 aa3t1 aa4t1 aa1t2 aa2t2 aa3t2 aa4t2 -------- -------- -------- -------- -------- -------- -------- -------- aa1t1 - - aa2t1 -0.57 - - aa3t1 0.53 -0.52 - - aa4t1 -0.20 2.07 -1.58 - - aa1t2 3.84 2.17 0.00 -0.90 - - aa2t2 -1.13 0.16 -0.14 -1.16 -0.49 - - aa3t2 -1.26 -1.89 4.90 0.81 -1.63 0.26 - - aa4t2 -0.74 -2.71 -0.10 0.41 -0.45 1.42 0.71 - -


Modification Indices and Expected Change Modification Indices for LAMBDA-X AAT1 AAT2 -------- -------- aa1t1 - - 0.11 aa2t1 - - 1.19 aa3t1 - - 2.48 aa4t1 - - 0.30 aa1t2 5.94 - - aa2t2 1.46 - - aa3t2 0.37 - - aa4t2 2.78 - -


Modification Indices for THETA-DELTA aa1t1 aa2t1 aa3t1 aa4t1 aa1t2 aa2t2 aa3t2 aa4t2 -------- -------- -------- -------- -------- -------- -------- -------- aa1t1 - - aa2t1 0.32 - - aa3t1 0.28 0.27 - - aa4t1 0.04 4.28 2.49 - - aa1t2 16.26 5.45 2.37 4.97 - - aa2t2 1.66 1.83 0.57 0.65 0.24 - - aa3t2 4.06 6.09 27.88 0.95 2.67 0.07 - - aa4t2 0.30 6.31 0.27 2.77 0.20 2.02 0.51 - -


TitleConfirmatory factor model (attitude toward using coupons measured at

two points in time) Observed Variables id aa1t1 aa2t1 aa3t1 aa4t1 aa1t2 aa2t2 aa3t2 aa4t2 Raw Data from File=d:\m554\eden2\cfa.dat Latent Variables AAT1 AAT2 Sample Size 250 Relationships aa1t1 aa2t1 aa3t1 aa4t1 = AAT1 aa1t2 aa2t2 aa3t2 aa4t2 = AAT2 Set the Variance of AAT1 AAT2 to 1 Set the Error Covariance of aa1t1 and aa1t2 free Set the Error Covariance of aa2t1 and aa2t2 free Set the Error Covariance of aa3t1 and aa3t2 free Set the Error Covariance of aa4t1 and aa4t2 free Options sc rs mi wp Path Diagram End of Problem

Confirmatory factor analysisGoodness of Fit Statistics: Degrees of Freedom = 15 Minimum Fit Function Chi-Square = 26.76 (P = 0.031) Normal Theory Weighted Least Squares Chi-Square = 26.28 (P = 0.035) Estimated Non-centrality Parameter (NCP) = 11.28 90 Percent Confidence Interval for NCP = (0.78 ; 29.60) Minimum Fit Function Value = 0.11 Population Discrepancy Function Value (F0) = 0.045 90 Percent Confidence Interval for F0 = (0.0031 ; 0.12) Root Mean Square Error of Approximation (RMSEA) = 0.055 90 Percent Confidence Interval for RMSEA = (0.014 ; 0.089) P-Value for Test of Close Fit (RMSEA < 0.05) = 0.37 Expected Cross-Validation Index (ECVI) = 0.27 90 Percent Confidence Interval for ECVI = (0.23 ; 0.35) ECVI for Saturated Model = 0.29 ECVI for Independence Model = 12.05 Chi-Square for Independence Model with 28 Degrees of Freedom = 2983.93 Independence AIC = 2999.93 Model AIC = 68.28 Saturated AIC = 72.00 Independence CAIC = 3036.10 Model CAIC = 163.23 Saturated CAIC = 234.77 Normed Fit Index (NFI) = 0.99 Non-Normed Fit Index (NNFI) = 0.99 Parsimony Normed Fit Index (PNFI) = 0.53 Comparative Fit Index (CFI) = 1.00 Incremental Fit Index (IFI) = 1.00 Relative Fit Index (RFI) = 0.98 Critical N (CN) = 285.55 Root Mean Square Residual (RMR) = 0.031 Standardized RMR = 0.017 Goodness of Fit Index (GFI) = 0.97 Adjusted Goodness of Fit Index (AGFI) = 0.94 Parsimony Goodness of Fit Index (PGFI) = 0.41


Completely Standardized Solution LAMBDA-X AAT1 AAT2 -------- -------- aa1t1 0.80 - - aa2t1 0.85 - - aa3t1 0.73 - - aa4t1 0.86 - - aa1t2 - - 0.85 aa2t2 - - 0.88 aa3t2 - - 0.79 aa4t2 - - 0.87 PHI AAT1 AAT2 -------- -------- AAT1 1.00 AAT2 0.89 1.00 THETA-DELTA aa1t1 aa2t1 aa3t1 aa4t1 aa1t2 aa2t2 aa3t2 aa4t2

-------- -------- -------- -------- -------- -------- -------- -------- aa1t1 0.36 aa2t1 - - 0.28 aa3t1 - - - - 0.47 aa4t1 - - - - - - 0.27 aa1t2 0.09 [.28] - - - - - - 0.29 aa2t2 - - 0.02 - - - - - - 0.22 aa3t2 - - - - 0.15 [.36] - - - - - - 0.37 aa4t2 - - - - - - 0.02 - - - - - - 0.24


CFA results

Construct ParameterParameterestimate

z-value ofparameter estimate

Individual-itemreliability

Composite reliability(average variance

extracted)

AAT1 .88 (.66)

λ111.08 14.78 0.64

λ211.11 16.07 0.72

λ310.92 12.98 0.53

λ411.22 16.33 0.73

θ110.66 9.12 --

θ220.48 8.06 --

θ330.76 9.88 --

θ440.55 7.87 --


Construct ParameterParameterestimate

z-value ofparameter estimate

Individual-itemreliability

Composite reliability(average variance

extracted)

AAT2 .91 (.72)

λ521.19 16.25 0.71

λ621.17 17.29 0.78

λ720.98 14.82 0.63

λ821.24 17.02 0.76

θ550.56 8.81 --

θ660.40 7.77 --

θ770.56 9.58 --

θ880.48 8.05 --

CFA results


Factor correlations

Original correlation Corrected correlation

Exploratory factor analysis (PFA with Promax rotation)

.75 n.a.

Confirmatory factor analysis .90 .90

Correlation of unweighted linear composites at t1, t2

.82

Average correlation of individual t1, t2 measures .63

91.911.882.

819.

91.719.654.

626.

confirmatory factor analysis

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