G89.2247 Lecture 101 SEM methods revisited Multilevel models revisited Multilevel models as...
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G89.2247 Lecture 10 1
G89.2247Lecture 10
• SEM methods revisited
• Multilevel models revisited
• Multilevel models as represented in SEM
• Examples
G89.2247 Lecture 10 2
SEM Method Reviewed
• Last week we considered a regressed change model
V2 V3 V5V4
V1
F1 F2
D2
E2 E3 E4 E5
G89.2247 Lecture 10 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 = 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
G89.2247 Lecture 10 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.
V2 V3 V5V4
V1
F1 F2 D2
E2 E3 E4 E5
F3
D1
D3
G89.2247 Lecture 10 5
SEM can also handle intercept terms
V2 V3 V5V4
V1
F1 F2 D2
E2 E3 E4 E5
F3
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
G89.2247 Lecture 10 6
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
G89.2247 Lecture 10 7
SEM systems of equations can be used for multilevel models
• Recall from Lecture 6, Level 1 and Level 2 EquationsE.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 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 = 00 + U0j
• B1j = 10 + U1j
G89.2247 Lecture 10 9
Level 1 Models in SEM
X1 X2 X4X3
B0 B1 U2
r1 r2 r3 r4
U11 1 1 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.
G89.2247 Lecture 10 10
Level 2 Equations in SEM
• This picture makes it clear that the intercept and slope are variables that reflect individual differences.
B0 B1 U2U1
1Group
G89.2247 Lecture 10 11
Full Model
X1 X2 X4X3
B0 B1 U2
r1 r2 r3 r4
U1
1Group
1 1 1 1 0 1 2 3
G89.2247 Lecture 10 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 = *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 13
Special Features of SEM Approach
• The Variances of r1, r2, r3 and r4 can be estimated separatelyLike PROC MIXED, they can also be constrained
to be the sameDefault is for heteroscedascity
• More than one set of slopes and intercepts can be examinedStructural relations of these trajectories can be
examined
G89.2247 Lecture 10 14
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 <.0001group -0.5742 0.1076 270 -5.34 <.0001week 0.2706 0.02428 133 11.14 <.0001group*week -0.2942 0.03446 270 -8.54 <.0001
Residual 0.1049 0.009032
PROC MIXED Results, no correlated residuals
G89.2247 Lecture 10 15
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
G89.2247 Lecture 10 16
Example: Anxiety over Weeks:Latent Growth Model via EQS
• Variances and Covariances E1 -SAMPLE .252*I D1 - F1 .314*I .031 I .048 I I I E2 -WEEK1 .106*I D2 - F2 .019*I .009 I .005 I I I E3 -WEEK2 .106*I I .009 I I I I E4 -WEEK3 .106*I I .009 I I I I E5 -WEEK4 .106*I I .009 I I Covariance of intercept and slope I D2 - F2 .008*I I D1 - F1 .011 I
G89.2247 Lecture 10 17
A Heteroscedascity Model GOODNESS OF FIT SUMMARY INDEPENDENCE MODEL CHI-SQUARE = 640.966 ON 10 DEGREES OF FREEDOM INDEPENDENCE AIC = 620.96566 INDEPENDENCE CAIC = 581.91291 MODEL AIC = 11.30153 MODEL CAIC = -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= 0.961 BENTLER-BONETT NONNORMED FIT INDEX= 0.959 COMPARATIVE FIT INDEX (CFI) = 0.971
Test of homoscedascity
26.7 (10df) – 25.3 (7df) = 1.4 (3df) [do not reject null]
G89.2247 Lecture 10 18
Variance Estimates
• One can see the variances are quite similar
E1 -SAMPLE .252*I D1 - F1 .312*I .031 I .049 I I I E2 -WEEK1 .111*I D2 - F2 .020*I .027 I .006 I I I E3 -WEEK2 .111*I I .018 I I I I E4 -WEEK3 .114*I I .019 I I I I E5 -WEEK4 .077*I I .027 I I
G89.2247 Lecture 10 19
A Correlated Error Model/EQUATIONSV1 = *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;/VARIANCESV999= 1;E1 = 10*;E2 = 10*;E3 = 10*;E4 = 10*;E5 = 10*;D1 = 10*;D2 = 10*;/COVARIANCESD2 , D1 = *;E2 , E3 = *;E3 , E4 = *;E4 , E5 = *;/CONSTRAINTS(E2,E2)=(E3,E3)=(E4,E4)=(E5,E5);(E2,E3)=(E3,E4)=(E4,E5);
G89.2247 Lecture 10 20
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
G89.2247 Lecture 10 21
Estimates from Correlated Residual ModelLevel 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
G89.2247 Lecture 10 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
/EQUATIONSV1 = *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;
G89.2247 Lecture 10 23
Results from Flex Time
• The improvement in Chi Square was nsSAMPLE =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
G89.2247 Lecture 10 24
Closing Remarks
• Latent Growth Models are an interesting alternative to Proc Mixed/HLM
• AdvantagesFlexible modeling featuresTruly multivariateMeasurement models could be incorporated
• Possible disadvantagesMissing data presents more complicationsNumber of time points may be limitedEmphasizes trajectories rather than process
• Active statistical work affects the balance of advantages and disadvantages