The general structural equation model with latent variates Hans Baumgartner Penn State University The general structural equation model Measurement model (Appendix A) The general structural equation model Latent variable model (Appendix A) The general structural equation model Why is it important to take into account measurement error? Consider the following model: Correction for attenuation: The general structural equation model Why is it important to take into account measurement error? (contd) The true effect of on is: The effect of x on y is: The general structural equation model Model identification Necessary condition for identification Two-step rule: Measurement model Latent variable model Null B rule Recursive rule Rank condition Empirical identification tests The general structural equation model Title A general structural equation model (explaining coupon usage) Observed Variables id be1 be2 be3 be4 be5 be6 be7 aa1t1 aa2t1 aa3t1 aa4t1 bi1 bi2 bh1 Raw Data from File=d:\m554\eden2\sem.dat Latent Variables INCONV REWARDS ENCUMBR AACT BI BH Sample Size 250 Relationships be1 = 1*INCONV be2 = INCONV be3 = 1*REWARDS be4 = REWARDS be5 = 1*ENCUMBR be6 = ENCUMBR be7 = ENCUMBR aa1t1 = 1*AACT aa2t1 = AACT aa3t1 = AACT aa4t1 = AACT bi1 = 1*BI bi2 = BI bh1 = 1*BH AACT = INCONV REWARDS ENCUMBR BI = AACT BH = BI Set the Error Variance of bh1 to zero Options sc rs mi wp Path Diagram End of Problem SIMPLEX specification The general structural equation model Goodness of Fit Statistics: Degrees of Freedom = 70 Minimum Fit Function Chi-Square = (P = 0.031) Normal Theory Weighted Least Squares Chi-Square = (P = 0.037) Estimated Non-centrality Parameter (NCP) = Percent Confidence Interval for NCP = (1.60 ; 51.68) Minimum Fit Function Value = 0.38 Population Discrepancy Function Value (F0) = Percent Confidence Interval for F0 = ( ; 0.21) Root Mean Square Error of Approximation (RMSEA) = Percent Confidence Interval for RMSEA = ( ; 0.054) P-Value for Test of Close Fit (RMSEA < 0.05) = 0.89 Expected Cross-Validation Index (ECVI) = Percent Confidence Interval for ECVI = (0.57 ; 0.77) ECVI for Saturated Model = 0.84 ECVI for Independence Model = Chi-Square for Independence Model with 91 Degrees of Freedom = Independence AIC = Model AIC = Saturated AIC = Independence CAIC = Model CAIC = Saturated CAIC = Normed Fit Index (NFI) = 0.97 Non-Normed Fit Index (NNFI) = 0.99 Parsimony Normed Fit Index (PNFI) = 0.75 Comparative Fit Index (CFI) = 0.99 Incremental Fit Index (IFI) = 0.99 Relative Fit Index (RFI) = 0.96 Critical N (CN) = Root Mean Square Residual (RMR) = 0.13 Standardized RMR = Goodness of Fit Index (GFI) = 0.95 Adjusted Goodness of Fit Index (AGFI) = 0.92 Parsimony Goodness of Fit Index (PGFI) = 0.63 The general structural equation model Measurement Equations aa1t1 = 1.00*AACT, Errorvar.= 0.68, R 2 = 0.63 (0.075) 9.06 aa2t1 = 1.04*AACT, Errorvar.= 0.44, R 2 = 0.74 (0.069) (0.058) aa3t1 = 0.85*AACT, Errorvar.= 0.76, R 2 = 0.53 (0.070) (0.077) aa4t1 = 1.10*AACT, Errorvar.= 0.59, R 2 = 0.71 (0.076) (0.072) bi1 = 1.00*BI, Errorvar.= 0.97, R 2 = 0.75 (0.14) 7.04 bi2 = 1.09*BI, Errorvar.= 0.25, R 2 = 0.93 (0.058) (0.13) bh1 = 1.00*BH,, R 2 = 1.00 The general structural equation model Measurement Equations be1 = 1.00*INCONV, Errorvar.= 0.56, R 2 = 0.79 (0.17) 3.32 be2 = 0.98*INCONV, Errorvar.= 0.61, R 2 = 0.77 (0.087) (0.16) be3 = 1.00*REWARDS, Errorvar.= 0.45, R 2 = 0.75 (0.18) 2.55 be4 = 0.82*REWARDS, Errorvar.= 0.96, R 2 = 0.48 (0.12) (0.15) be5 = 1.00*ENCUMBR, Errorvar.= 2.78, R 2 = 0.24 (0.28) 9.97 be6 = 1.73*ENCUMBR, Errorvar.= 1.85, R 2 = 0.59 (0.27) (0.34) be7 = 1.48*ENCUMBR, Errorvar.= 1.92, R 2 = 0.50 (0.24) (0.28) The general structural equation model Structural Equations AACT = *INCONV *REWARDS *ENCUMBR, Errorvar.= 0.69, R 2 = 0.42 (0.058) (0.081) (0.097) (0.11) BI = 1.10*AACT, Errorvar.= 1.53, R 2 = 0.48 (0.11) (0.20) BH = 0.49*BI, Errorvar.= 1.41, R 2 = 0.34 (0.049) (0.13) The general structural equation model Modification Indices for BETA AACT BI BH AACT BI BH Modification Indices for GAMMA INCONV REWARDS ENCUMBR AACT BI BH