G89 2247 Lecture 10 SEM methods revisited Multilevel models revisited Multilevel models as represented in SEM Examples G89 2247 Lecture 10 1 SEM Method Reviewed Last week we considered a regressed change model V1 D2 F1 F2 V2 V3 V4 V5 E2 E3 E4 E5 G89 2247 Lecture 10 2 EQS Equations Lord s Paradox Example Equations involving Latent Variables SEPTA V2 1 000 F1 1 000 E2 SEPTB V3 1 017 F1 1 000 E3 V4 1 000 F2 1 000 E4 MAYA F1 MAYB V5 1 000 E5 F2 are factors 1 012 F2 indicates estimates Estimates based on Covariance Structure of V1 V5 F2 F2 11 164 V1 1 000 D2 Results suggest modest group effect 749 F1 on regressed change G89 2247 Lecture 10 3 No Change All Selection We considered an alternative model that suggested that group effects were the same at both times This model has same fit V1 D3 F3 F1 D1 F2 D2 V2 V3 V4 V5 E2 E3 E4 E5 G89 2247 Lecture 10 4 SEM can also handle intercept terms The triangle shows the effect of a constant intercept on variable values In this model the constant works toward V2 V5 through the latent variables 1 V1 F3 D3 F1 D1 F2 D2 V2 V3 V4 V5 E2 E3 E4 E5 G89 2247 Lecture 10 5 EQS Equations for Constant Model GROUP V1 SEPTA V2 SEPTB V3 MAYA 500 V999 1 000 E1 1 000 F1 1 000 E2 998 F1 1 000 E3 V4 1 000 F2 1 000 E4 MAYB V5 1 003 F2 1 000 E5 F3 F3 41 782 V1 132 143 V999 F1 F1 1 000 F3 1 000 D1 F2 F2 1 000 F3 1 000 D2 1 000 D3 V999 is the constant term in EQS F3 is 132 for females and 174 for males The replicate measures in each month give close results G89 2247 Lecture 10 6 SEM systems of equations can be used for multilevel models Recall from Lecture 6 Level 1 and Level 2 Equations E g linear change over four times Suppose Yij is an outcome and Xj contains codes for time Xj 0 1 2 3 Level 1 equation Yij B0j B1jXj rij Level 2 equations B0j 00 U0j B1j 10 U1j G89 2247 Lecture 10 7 Systems of Equations continued Spelling out level 1 equations for X ij 0 1 2 3 Y1j B0j B1j0 rij Y2j B0j B1j1 rij Y3j B0j B1j2 rij Y4j B0j B1j3 rij Level 2 equations B0j 00 U0j B1j 10 U1j G89 2247 Lecture 10 8 Level 1 Models in SEM Diagram looks like confirmatory factor analysis but the loading are fixed not estimated Within person processes are inferred from between person covariance patterns B0 U1 1 1 B1 1 1 0 U2 1 2 3 X1 X2 X3 X4 r1 r2 r3 r4 G89 2247 Lecture 10 9 Level 2 Equations in SEM Group 1 U1 B0 B1 U2 This picture makes it clear that the intercept and slope are variables that reflect individual differences G89 2247 Lecture 10 10 Full Model Group 1 B0 U1 1 1 B1 1 1 0 U2 1 2 3 X1 X2 X3 X4 r1 r2 r3 r4 G89 2247 Lecture 10 11 Model as EQS Equations EQUATIONS V1 V999 E1 V2 1F1 0F2 E2 V3 1F1 1F2 E3 V4 1F1 2F2 E4 V5 1F1 3F2 E5 F1 V999 V1 D1 F2 V999 V1 D2 VARIANCES V999 1 E1 10 E2 10 E3 10 E4 10 E5 10 D1 10 D2 10 COVARIANCES D2 D1 0 CONSTRAINTS E2 E2 E3 E3 E4 E4 E5 E5 G89 2247 Lecture 10 12 Special Features of SEM Approach The Variances of r1 r2 r3 and r4 can be estimated separately Like PROC MIXED they can also be constrained to be the same Default is for heteroscedascity More than one set of slopes and intercepts can be examined Structural relations of these trajectories can be examined G89 2247 Lecture 10 13 Example Anxiety over Weeks PROC MIXED Results no correlated residuals Estimated G Matrix Row 1 2 Effect Intercept week id 1 1 Col1 0 3175 0 007463 Col2 0 007463 0 01909 Estimated G Correlation Matrix Row 1 2 Effect Intercept week id 1 1 Col1 1 0000 0 09586 Col2 0 09586 1 0000 Solution for Fixed Effects Effect Intercept group week group week Residual Estimate 1 1276 0 5742 0 2706 0 2942 S Error 0 07583 0 1076 0 02428 0 03446 0 1049 0 009032 DF 133 270 133 270 G89 2247 Lecture 10 t Value 14 87 5 34 11 14 8 54 Pr t 0001 0001 0001 0001 14 Example Anxiety over Weeks Latent Growth Model via EQS GOODNESS OF FIT SUMMARY CHI SQUARE 26 679 BASED ON 10 DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI SQUARE STATISTIC IS 0 00293 BENTLER BONETT NORMED FIT INDEX 0 958 BENTLER BONETT NONNORMED FIT INDEX 0 974 COMPARATIVE FIT INDEX CFI 0 974 SAMPLE V1 496 V999 043 1 000 E1 WEEK1 V2 1 000 F1 1 000 E2 WEEK2 WEEK3 WEEK4 V3 V4 V5 1 000 F1 1 000 F1 1 000 F1 1 000 F2 2 000 F2 3 000 F2 1 000 E3 1 000 E4 1 000 E5 F1 F1 F2 1 128 V999 076 271 V999 024 1 000 D1 F2 575 V1 107 294 V1 034 G89 2247 Lecture 10 1 000 D2 15 Example Anxiety over Weeks Latent Growth Model via EQS Variances and Covariances E1 SAMPLE E2 WEEK1 E3 WEEK2 E4 WEEK3 E5 WEEK4 252 I D1 031 I I 106 I D2 009 I I 106 I 009 I I 106 I 009 I I 106 I 009 I F1 F2 314 I 048 I I 019 I 005 I I I I I I I I I I Covariance of intercept and slope I D2 F2 008 I I D1 F1 011 I G89 2247 Lecture 10 16 A Heteroscedascity Model GOODNESS OF FIT SUMMARY INDEPENDENCE MODEL CHI SQUARE FREEDOM INDEPENDENCE AIC MODEL AIC 620 96566 11 30153 640 966 ON INDEPENDENCE CAIC MODEL CAIC 10 DEGREES OF 581 91291 16 03539 CHI SQUARE 25 302 BASED ON 7 DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI SQUARE STATISTIC IS LESS THAN 0 001 THE NORMAL THEORY RLS CHI SQUARE FOR THIS ML SOLUTION IS 24 702 BENTLER BONETT NORMED FIT INDEX BENTLER BONETT NONNORMED FIT INDEX COMPARATIVE FIT INDEX CFI 0 961 0 959 0 971 Test of homoscedascity 26 7 10df 25 3 7df 1 4 3df do not reject null G89 2247 Lecture 10 17 Variance Estimates One can see the variances are quite similar E1 SAMPLE E2 WEEK1 E3 WEEK2 E4 WEEK3 E5 WEEK4 252 I D1 031 I I 111 I D2 027 I I 111 I 018 I I 114 I 019 I I 077 I 027 I F1 F2 G89 2247 Lecture 10 312 I 049 I I 020 I 006 I I I I I I I I I I 18 A Correlated Error Model EQUATIONS V1 V999 E1 V2 1F1 0F2 E2 …