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**G89.2247 Lecture 10 SEM methods revisited Multilevel models revisited**

Multilevel models as represented in SEM Examples G Lecture 10

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**SEM Method Reviewed Last week we considered a regressed change model**

F1 F2 V4 V2 V3 V5 E4 E5 E2 E3 G Lecture 10

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**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 = F E2 SEPTB =V3 = *F E3 MAYA =V4 = F E4 MAYB =V5 = *F E5 F2 =F2 = *V *F D2 G Lecture 10

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**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. D3 V1 F3 F1 D1 F2 D2 V4 V2 V3 V5 E4 E5 E2 E3 G Lecture 10

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**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 D3 V1 F3 F1 F2 D2 D1 V4 V2 V3 V5 E4 E5 E2 E3 G Lecture 10

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**EQS Equations for Constant Model**

GROUP =V1 = *V E1 SEPTA =V2 = F E2 SEPTB =V3 = *F E3 MAYA =V4 = F E4 MAYB =V5 = *F E5 F3 =F3 = *V *V D3 F1 =F1 = F D1 F2 =F2 = *F D2 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 G Lecture 10

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**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 = g00 + U0j B1j = g10 + U1j G Lecture 10

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**Systems of Equations, continued**

Spelling out level 1 equations for Xij =0,1,2,3 Y1j = B0j + B1j0 + rij Y2j = B0j + B1j1 + rij Y3j = B0j + B1j2 + rij Y4j = B0j + B1j3 + rij Level 2 equations B0j = g00 + U0j B1j = g10 + U1j G Lecture 10

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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 B1 U1 U2 1 2 3 1 X1 X2 X3 X4 r1 r2 r3 r4 G Lecture 10

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Level 2 Equations in SEM Group 1 B0 B1 U1 U2 This picture makes it clear that the intercept and slope are variables that reflect individual differences. G Lecture 10

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**Full Model Group 1 B0 B1 U1 U2 X1 X2 X3 X4 r1 r2 r3 r4 1 2 3 1**

1 2 3 1 X1 X2 X3 X4 r1 r2 r3 r4 G Lecture 10

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**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 = *V *V1 + D1; F2 = *V *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); G Lecture 10

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**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 G Lecture 10

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**Example: Anxiety over Weeks**

PROC MIXED Results, no correlated residuals Estimated G Matrix Row Effect id Col Col2 1 Intercept 2 week Estimated G Correlation Matrix 1 Intercept 2 week Solution for Fixed Effects Effect Estimate S. Error DF t Value Pr > |t| Intercept <.0001 group <.0001 week <.0001 group*week <.0001 Residual G Lecture 10

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**Example: Anxiety over Weeks: Latent Growth Model via EQS**

GOODNESS OF FIT SUMMARY CHI-SQUARE = BASED ON DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS BENTLER-BONETT NORMED FIT INDEX= BENTLER-BONETT NONNORMED FIT INDEX= COMPARATIVE FIT INDEX (CFI) = SAMPLE =V1 = *V E1 .043 WEEK1 =V2 = F E2 WEEK2 =V3 = F F E3 WEEK3 =V4 = F F E4 WEEK4 =V5 = F F E5 F1 =F1 = *V *V D1 F2 =F2 = *V *V D2 G Lecture 10

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**Example: Anxiety over Weeks: Latent Growth Model via EQS**

Variances and Covariances G Lecture 10

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**A Heteroscedascity Model**

Test of homoscedascity 26.7 (10df) – 25.3 (7df) = 1.4 (3df) [do not reject null] G Lecture 10

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**Variance Estimates One can see the variances are quite similar**

G Lecture 10

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**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 = *V *V1 + D1; F2 = *V *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); G Lecture 10

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**Results from Correlated Error Model**

GOODNESS OF FIT SUMMARY INDEPENDENCE MODEL CHI-SQUARE = ON DEGREES OF FREEDOM CHI-SQUARE = BASED ON DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS Test of Correlated Errors 26.7 (10df) – 15.4 (9df) = 11.3 (1df) Significant G Lecture 10

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**Estimates from Correlated Residual Model**

Level 2 equations and estimates (Fixed Effects) SAMPLE =V1 = *V E1 .043 F1 =F1 = *V *V D1 F2 =F2 = *V *V D2 Correlations of Effects E3 -WEEK *I D F *I E2 -WEEK I D F I I I E4 -WEEK *I I E3 -WEEK I I E5 -WEEK *I I E4 -WEEK I I G Lecture 10

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**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; G Lecture 10

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**Results from Flex Time The improvement in Chi Square was ns**

SAMPLE =V1 = *V E1 .043 WEEK1 =V2 = F E2 WEEK2 =V3 = F *F E3 .171 WEEK3 =V4 = F *F E4 .165 WEEK4 =V5 = F F E5 G Lecture 10

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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 G Lecture 10

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