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