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G89.2247 Lecture 101 SEM methods revisited Multilevel models revisited Multilevel models as represented in SEM Examples

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G89.2247 Lecture 102 SEM Method Reviewed Last week we considered a regressed change model V2V3V5 V4 V1 F1F1 F2F2 D2 E2E3 E4E5

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G89.2247 Lecture 103 EQS Equations (Lord's Paradox Example) Equations involving Latent Variables F1, F2 are factors, * indicates estimates Estimates based on Covariance Structure of V1—V5 Results suggest modest group effect on regressed change SEPTA =V2 = 1.000 F1 + 1.000 E2 SEPTB =V3 = 1.017*F1 + 1.000 E3 MAYA =V4 = 1.000 F2 + 1.000 E4 MAYB =V5 = 1.012*F2 + 1.000 E5 F2 =F2 = 11.164*V1 +.749*F1 + 1.000 D2

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G89.2247 Lecture 104 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. V2V3V5 V4 V1 F1F1 F2F2 D2 E2E3 E4E5 F3F3 D1 D3

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G89.2247 Lecture 105 SEM can also handle intercept terms V2V3V5 V4 V1 F1F1 F2F2 D2 E2E3 E4E5 F3F3 D1 1 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. D3

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G89.2247 Lecture 106 EQS Equations for Constant Model 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 GROUP =V1 =.500*V999 + 1.000 E1 SEPTA =V2 = 1.000 F1 + 1.000 E2 SEPTB =V3 =.998*F1 + 1.000 E3 MAYA =V4 = 1.000 F2 + 1.000 E4 MAYB =V5 = 1.003*F2 + 1.000 E5 F3 =F3 = 41.782*V1 +132.143*V999 + 1.000 D3 F1 =F1 = 1.000 F3 + 1.000 D1 F2 =F2 = 1.000*F3 + 1.000 D2

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G89.2247 Lecture 107 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 Y ij is an outcome and X j contains codes for time (X j =0,1,2,3) Level 1 equation Y ij = B 0j + B 1j X j + r ij Level 2 equations B 0j = 00 + U 0j B 1j = 10 + U 1j

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G89.2247 Lecture 108 Systems of Equations, continued Spelling out level 1 equations for X ij =0,1,2,3 Y 1j = B 0j + B 1j 0 + r ij Y 2j = B 0j + B 1j 1 + r ij Y 3j = B 0j + B 1j 2 + r ij Y 4j = B 0j + B 1j 3 + r ij Level 2 equations B 0j = 00 + U 0j B 1j = 10 + U 1j

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G89.2247 Lecture 109 Level 1 Models in SEM X1X2 X4 X3 B0B0 B1B1 U2 r1r2r3r4 U1 1 11 1 0 1 2 3 Diagram looks like confirmatory factor analysis, but the "loading" are fixed, not estimated. Within person processes are inferred from between person covariance patterns.

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G89.2247 Lecture 1010 Level 2 Equations in SEM This picture makes it clear that the intercept and slope are variables that reflect individual differences. B0B0 B1B1 U2U1 1 Group

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G89.2247 Lecture 1011 Full Model X1X2 X4 X3 B0B0 B1B1 U2 r1r2r3r4 U1 1 Group 1 11 1 0 1 2 3

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G89.2247 Lecture 1012 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);

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G89.2247 Lecture 1013 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

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G89.2247 Lecture 1014 Example: Anxiety over Weeks Estimated G Matrix Row Effect id Col1 Col2 1 Intercept 1 0.3175 0.007463 2 week 1 0.007463 0.01909 Estimated G Correlation Matrix Row Effect id Col1 Col2 1 Intercept 1 1.0000 0.09586 2 week 1 0.09586 1.0000 Solution for Fixed Effects Effect Estimate S. Error DF t Value Pr > |t| Intercept 1.1276 0.07583 133 14.87 <.0001 group -0.5742 0.1076 270 -5.34 <.0001 week 0.2706 0.02428 133 11.14 <.0001 group*week -0.2942 0.03446 270 -8.54 <.0001 Residual 0.1049 0.009032 PROC MIXED Results, no correlated residuals

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G89.2247 Lecture 1015 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 + 1.000 E1.043 WEEK1 =V2 = 1.000 F1 + 1.000 E2 WEEK2 =V3 = 1.000 F1 + 1.000 F2 + 1.000 E3 WEEK3 =V4 = 1.000 F1 + 2.000 F2 + 1.000 E4 WEEK4 =V5 = 1.000 F1 + 3.000 F2 + 1.000 E5 F1 =F1 = -.575*V1 + 1.128*V999 + 1.000 D1.107.076 F2 =F2 = -.294*V1 +.271*V999 + 1.000 D2.034.024

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G89.2247 Lecture 1016 Example: Anxiety over Weeks: Latent Growth Model via EQS Variances and Covariances

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G89.2247 Lecture 1017 A Heteroscedascity Model Test of homoscedascity 26.7 (10df) – 25.3 (7df) = 1.4 (3df) [do not reject null]

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G89.2247 Lecture 1018 Variance Estimates One can see the variances are quite similar

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G89.2247 Lecture 1019 A Correlated Error Model /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 = *; E2, E3 = *; E3, E4 = *; E4, E5 = *; /CONSTRAINTS (E2,E2)=(E3,E3)=(E4,E4)=(E5,E5); (E2,E3)=(E3,E4)=(E4,E5);

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G89.2247 Lecture 1020 Results from Correlated Error Model GOODNESS OF FIT SUMMARY INDEPENDENCE MODEL CHI-SQUARE = 640.966 ON 10 DEGREES OF FREEDOM CHI-SQUARE = 15.361 BASED ON 9 DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS 0.08149 Test of Correlated Errors 26.7 (10df) – 15.4 (9df) = 11.3 (1df) Significant

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G89.2247 Lecture 1021 Estimates from Correlated Residual Model Level 2 equations and estimates (Fixed Effects) SAMPLE =V1 =.496*V999 + 1.000 E1.043 F1 =F1 = -.599*V1 + 1.149*V999 + 1.000 D1.106.075 F2 =F2 = -.284*V1 +.263*V999 + 1.000 D2.034.024 Correlations of Effects E3 -WEEK2.298*I D2 - F2.754*I E2 -WEEK1 I D1 - F1 I I I E4 -WEEK3.298*I I E3 -WEEK2 I I I I E5 -WEEK4.298*I I E4 -WEEK3 I I

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G89.2247 Lecture 1022 A Model for Flexible Time Suppose that psychological time to event is not perfectly mapped on weekly time. We can relax the time structure to see if different weights are better in estimating trajectories /EQUATIONS V1 = *V999 + E1; V2 = + 1F1 + 0F2 + E2; V3 = + 1F1 + 1*F2 + E3; V4 = + 1F1 + 2*F2 + E4; V5 = + 1F1 + 3F2 + E5; F1 = *V999 + *V1 + D1; F2 = *V999 + *V1 + D2;

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G89.2247 Lecture 1023 Results from Flex Time The improvement in Chi Square was ns SAMPLE =V1 =.496*V999 + 1.000 E1.043 WEEK1 =V2 = 1.000 F1 + 1.000 E2 WEEK2 =V3 = 1.000 F1 +.619*F2 + 1.000 E3.171 WEEK3 =V4 = 1.000 F1 + 1.996*F2 + 1.000 E4.165 WEEK4 =V5 = 1.000 F1 + 3.000 F2 + 1.000 E5

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G89.2247 Lecture 1024 Closing Remarks Latent Growth Models are an interesting alternative to Proc Mixed/HLM Advantages Flexible modeling features Truly multivariate Measurement models could be incorporated Possible disadvantages Missing data presents more complications Number of time points may be limited Emphasizes trajectories rather than process Active statistical work affects the balance of advantages and disadvantages

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