1. G89.2247 Lecture 10 SEM methods revisited Multilevel models revisited
Multilevel models as represented in SEM
Examples
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2. SEM Method Reviewed Last week we considered a regressed change modelF1 F2 V4 V2 V3 V5 E4 E5 E2 E3
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3. 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
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4. 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
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5. 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
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6. 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
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7. 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
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8. 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
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9. 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
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10. 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.
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11. Full Model Group 1 B0 B1 U1 U2 X1 X2 X3 X4 r1 r2 r3 r4 1 2 3 11 2 3 1 X1 X2 X3 X4 r1 r2 r3 r4
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12. 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);
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13. 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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14. 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
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15. 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
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16. Example: Anxiety over Weeks: Latent Growth Model via EQS
Variances and Covariances
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17. A Heteroscedascity Model
Test of homoscedascity
26.7 (10df) – 25.3 (7df) = 1.4 (3df) [do not reject null]
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18. Variance Estimates One can see the variances are quite similar
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19. 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);
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20. 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
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21. 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
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22. 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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23. 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
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24. 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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