1. OLGA MITINA MOSCOW STATE UNIVERSITY
USING STRUCTURAL EQUATION MODELING
FOR NONLINEAR DYNAMIC SYSTEM THEORY
OLGA MITINA
MOSCOW STATE UNIVERSITY

2. STRUCTURAL EQUATION MODELING (SEM)
(Bentler, Jöreskog)
Confirmatory Factor Analysis (CFA)
Path Analysis
Growth Curve Modeling
Multilevel analysis
Multigroup Analysis

3. Basic Ideas of SEM Measured Variables ↔ Latent Variables Determination
↔ Correlation
↔ Variance E1 V1
Independent Variables
↔ Dependent Variables E2 V2 F1
Error (Residuals)
↔ Variable’s Specificity (Noise) E3 V3 V4 F2 D2 V5 F4 V6 F3 D3 E6

4. Basic Stages of SEM * Choosing measured variables E1 V1 Model Design *
0.8* 1 *
Choosing parameters:
Fixed parameters E2 V2 * F1
↔ Evaluated parameters * *
Data Collection E3 V3 *
V1=0.8*F1+ E1
V2= *F1+ E2
V3= *F1+ E3
V6= *V5+0.6F6+ E6
F2= *V4+1.0F4+ D2
F3=0.7*F D3
/VAR
E1=*; E2=*; E3=*; E6=*;
D2=*; D3=*; V4=*; V5=*;
F1=1; F4=*;
/COVAR
(V4,V5)=0.3;
(V4,F1)=*; (F1,F4)=*;
(E2,E3)=*; (E6,D3)=*; (D2,D3)=* * * V4 * F2
0.3 1 * * D2 V5 * F4 *
0.7* E6
0.6 V6 F3 * D3 D3 *

5. Basic Stages of SEM (Cont)
Calculating evaluated parameters
Calculating experimental matrix of covariations (correlations) for measured variables
Calculating theoretical matrix of covariations for measured variables
(on the basis of the model)
Modifying the values of evaluated parameters
Comparing theoretical and experimental matrices
Difference is large
Difference is Small
Statistical test of the Model

6. The Statement that theoretical model is adequate to experimental data is H0 Hypothesis
Using Statistics
Χ2 (df) – see tables
Χ2/df < 2
CFI (Comparative Fit Index) – 0<CFI≤1, should be >0.95
ROOT MEAN-SQUARE ERROR OF APPROXIMATION (RMSEA) -
0<RMSEA≤1, should be <0.05

7. NONLINEAR REGRESSION * * Measured Variables X Y=b0+b1*X+b2* Z+EY
+ b3*XZ
b1* *
b3* 1 EY Y ZX
b2* Z
Y=b0+b1*X+b2*X2+b3*X3+ EY X
b1* * 1
b2* EY Y X2
b3* X3

8. LINEAR REGRESSION Latent Variables F3=b0*+b1*F1 +b2*F2 + D3 E1 V1 V1

9. NONLINEAR REGRESSION Latent Variables E1 V1 V1 V1 V1 E2 V2 V2 F1 E3 V3
F3=b0*+b1*F1+b2*F2+b3*F1F2+D3 E7
V1V4 E8
V2V5
F1F2 F3 E9
V3V6 D3 E4 V4 E5 V5 F2 E6 V6

10. ONE-FACTOR MODEL V1 V2 F1 V3 Experimental data E1 E2 E3 “Error”
(ignored by SEM)
The Determined part of Measured Variable
(studied by SEM)
“Noise”
(could be studied by NDS)

11. Specific Effect (Newcomb)V1 V1 V1 V1 E2 V2 V2 F1 E3 V3 V3 E4 V4 V7 E7 E5 V5 F2 F3 V8 E8 E6 V6 V9 E9 D3

12. TRADITIONAL LINEAR RECURRENT MODEL FOR REPEATED MEASUREMENTSY1 Y2 Y3 …
Yn-1 Yn
Y2=λ12*Y1+E2
Y3= λ13*Y1+λ23*Y2+E3 …
Yn= λ1n*Y1+λ2n*Y2+… +λ2n*Yn-1 +En

13. NONLINEAR RECURRENT MODELY1 Y2 Y3 …
Yn-1 Yn
G11(Y1)
G22(Y2)
G33(Y3)
Gn-1n-1(Yn-1)
G12(Y1)
G23(Y2)
Gn-2n-1(Yn-2) .
G13(Y1)
G2n-1(Y2)
G1n-1(Y1)
Y2=Y1+G11(Y1)*+E2
Y3=Y2+G12(Y1)*+G22(Y2)*+E3 …
Yn= Yn-1+G1n-1(Y1)*+G2n-1(Y2)*+… +Gn-1n-1(Yn-1)* +En

14. TRADITIONAL LINEAR RECURRENT MODEL DETERMINANTS AND COVARIATES
WITH TIME INVARIANT
DETERMINANTS AND COVARIATES E1 E2 E3
En-1 En Y1 Y2 Y3 …
Yn-1 Yn ŵ
Y1=ā1* ŵ+E1
(Y1, ŵ)=*
Y2=λ11*Y1+ ā2* ŵ+E2
Y3= λ12*Y1+λ22*Y2+ ā3* ŵ +E3 …
Yn= λ1n-1*Y1+λ2n-1*Y2+… +λ2n-1*Yn-1 + ān* ŵ+En

15. WITH NONLINEAR TIME INVARIANT DETERMINANTS AND COVARIATES
RECURRENT MODEL
WITH NONLINEAR TIME INVARIANT
DETERMINANTS AND COVARIATES E1 E2 E3
En-1 En Y1 Y2 Y3 …
Yn-1 Yn ŵ
Q1(ŵ)
Q2(ŵ)
Q3(ŵ)
Qn-1(ŵ)
Qn(ŵ)
Y1= Q1(ŵ)*+E1
(Y1, Q1(ŵ))=*
Y2=λ11*Y1+ Q2(ŵ)*+E2
Y3= λ12*Y1+λ22*Y2+ Q3(ŵ)* +E3 …
Yn= λ1n-1*Y1+λ2n-1*Y2+… +λ2n-1*Yn-1 + Qn(ŵ)*+En

16. WITH NONLINEAR TIME INVARIANT DETERMINANTS AND COVARIATES
RECURRENT MODEL
WITH NONLINEAR TIME INVARIANT
DETERMINANTS AND COVARIATES E1 E2 E3
En-1 En Y1 Y2 Y3 …
Yn-1 Yn ŵ Û
Q1(Û)
Q2(Û)
Q3(Û)
Qn-1(Û)
Qn(Û)
Y1= Q1(Û)*+E1
Y2=λ11*Y1+ Q2(Û)*+E2
Y3= λ12*Y1+λ22*Y2+ Q3(Û)* +E3 …
Yn= λ1n-1*Y1+λ2n-1*Y2+… +λ2n-1*Yn-1 + Qn(Û)*+En

17. LATENT GROWTH CURVE MODELY1 Y2 Y3 …
Yn-1 Yn 1 1 1 1 1 1 1 2 2
n-1 n n 1 4
(n-1)2
(n-1)2 n2 n2
Constant
Linear
Quadratic
Cubic …
Level
Shape
Y1= 1F E1
Y2= 1F1+ 1F F F E2
Y3= 1F1+ 2F F F E3
Yn-1=1F1+(n-1)F2+(n-1)2F3+(n-1)3F4+ En-1
Yn= 1F1+ nF2+ n2F3+ n3F4+ En

18. LATENT GROWTH CURVE MODEL For analysis PolyDrug Use
(data were received from M.Newcomb and P.Bentler)
Age of respondents:
12-13
16-17
20-21
24-25
28-29
32-33 C1 A1 I1 C2 A2 I2 C3 A3 I3 C4 A4 I4 C5 A5 I5 C6 A6 I6
Polydrug Use1
Polydrug Use2
Polydrug Use3
Polydrug Use4
Polydrug Use5
Polydrug Use6
Constant
Linear
Quadratic
Cubic
CHI-SQUARE = 172
df = 93
CFI = .981
RMSEA = .051
-.050
.232
.045
-.265
.130
-.186
-.223
-.826
Gender
1-male, 2-female
F2 Academic Orientation
F3 Social Conformity
Yn= n n n3 + En

19. LATENT GROWTH CURVE MODEL WITH DISCONTINUITY EFFECT… Ym
Ym+1
Ym+2 … Yn
discontinuity
Level1
Shape1
Level2
Shape2

20. F2 Academic Orientation
CURVE-OF-FACTOR LGM
Age of respondents:
12-13
16-17
20-21
24-25
28-29
32-33 C1 A1 I1 C2 A2 I2 C3 A3 I3 C4 A4 I4 C5 A5 I5 C6 A6 I6
Polydrug Use1
Polydrug Use2
Polydrug Use3
Polydrug Use4
Polydrug Use5
Polydrug Use6
discontinuity
Level1
Shape1
Level2
Shape2
-.798
.336
.183
.239
-1.093
-.243
-.213
-.347
-.258
-1.514
Gender
1-male, 2-female
F1 Sensation Seeking
F2 Academic Orientation
F3 Social Conformity
Yn= n+En
CHI-SQUARE = 164
df = 93
CFI = .982
RMSEA = .048
Yn= n+En

21. THE NONLINEAR SINGLE-FACTOR SHAPE MODELX1 X2 X3 X4
X1Y1
X2Y2
X3Y3
X4Y4 Y1 Y2 Y3 Y4
Shape F1
Shape
F1F2
Shape F2
Level=0
Shape F3 D3
F3=b1F1+b2F2+b3F1F2+D3 Z1 Z2 Z3 Z4
E13
E14
E15
E16

22. Using Multilevel analysis for the study of self-organizing process
Individual Level
Political groups’ Level
Human rights should have priority over and above national interests.
Political
Liberalism
Every person should be allowed to leave his or her home country and return without any limitations.
Every person should be allowed to practice a religion of his or her choice without any limitations
General
Liberalism
Every person should be allowed to buy and sell land without any limitations.
Economical
Liberalism
Economy should evolve on the basis of  a free market system.
Commodity and financial circulation among countries should be allowed without limitations.

23. Acknowledgments
International Research and Exchange Board (IREX), the Bureau of Educational and Cultural Affairs,
US Department of State
Colleagues from the Laboratory headed by Prof. Peter Bentler, Psychology department, University of California, Los Angeles
Prof. Michael Newcomb from the University of South California