1. Multivariate Genetic Analysis
Boulder 2004

2. Phenotypic Cholesky T1 P2 F1 F4 P1 P3 P4 F2 F3
1.00

3. Phenotypic Cholesky T1 P2 F1 F4 P1 P3 P4 F2 F3
1.00

4. Phenotypic Cholesky T1 P2 F1 F4 P1 P3 P4 F2 F3
1.00

5. Phenotypic Cholesky T1 P2 F1 F4 P1 P3 P4 F2 F3
1.00

6. Cholesky Decomposition

7. Cholesky Example Script: f:\hmaes\a17\chol.mx
3 MRI measures – 2 IQ subtests:
var1 = cerebellum
var2 = grey matter
var3 = white matter
var4 = calculation
var5 = letters and numbers
Data: f:\hmaes\a17\mri-iqfactor.rec
T:\hmaes/a17\mri-iq-mz.dat
T:\hmaes\a17\mri-iq-dz.dat

8. Cholesky #define nvar 5 Begin Matrices;
X lower nvar nvar Free ! genetic structure
Y lower nvar nvar Free ! shared environment structure
Z lower nvar nvar Free ! specific environment structure
M full 1 nvar Free ! grand means
End Matrices;
Begin Algebra;
A= X*X'; ! additive genetic covariance
C= Y*Y'; ! shared environment covariance
E= Z*Z'; ! nonshared environment covariance
End Algebra;

9. Standardization Begin Algebra; R=A+C+E; ! total variance
S=(\sqrt(I.R))~; ! diagonal matrix of standard deviations
P=S*X| S*Y| S*Z; ! standardized estimates
End Algebra;

10. Phenotypic Single Factor
1.00

11. Residual Variances T1 P2 F1 P1 P3 P4
1.00 E1 E2 E3 E4

12. Twin Data T1 P2 F1 P1 P3 P4 T2 E1 E2 E3 E4
1.00 ?

13. Genetic Single Factor T1 P2 A1 P1 P3 P4 T2 E1 E2 E3 E4
1.00
1.0 / 0.5

14. Single [Common] Factor
X: genetic
Full 4 x 1
Full nvar x nfac
Y: shared environmental
Z: specific environmental

15. Common Environmental Single FactorP2 A1 P1 P3 P4 T2 E1 E2 E3 E4
1.00
1.0 / 0.5 C1

16. Specific Environmental Single Factor
1.00
1.0 / 0.5 C1

17. Residuals partitioned in ACE
1.00
1.0 / 0.5

18. Residual Factors T: genetic U: shared environmental
V: specific environmental
Diag 4 x 4
Diag nvar x nvar

19. Independent Pathway ModelC A E P1 P3 P4 T2 A1 A2 A3 A4 E1 E2 E3 E4 C1
1.00
1.00 / 0.50
[X]
[Y]
[Z]
[T]
[V]
[U]

20. Independent Pathway Example
Script: f:\hmaes\a17\ind3f.mx
3 MRI measures – 2 IQ subtests:
var1 = cerebellum
var2 = grey matter
var3 = white matter
var4 = calculation
var5 = letters and numbers
Data: f:\hmaes\a17\mri-iqfactor.rec
T:\hmaes/a17\mri-iq-mz.dat
T:\hmaes\a17\mri-iq-dz.dat

21. Independent Pathway #define nvar 5 #define nfac 1 G1: Define matrices
Calculation
Begin Matrices;
X full nvar ! common factor genetic paths
Y full nvar nfac ! common factor shared env paths
Z full nvar nfac Free ! common factor specific env paths
T diag nvar nvar Free ! variable specific genetic paths
U diag nvar nvar ! variable specific shared env paths
V diag nvar nvar Free ! variable specific residual paths
M full 1 nvar Free ! grand means
End Matrices;

22. Three Genetic Factors Specify X
! declare free parameters for 3 genetic common factors
! G1 G2 G3
Drop T 1 5 5
! fix genetic specific for last variable for identification

23. Three Genetic Factors CB GM WM RK CL
1.00 A1 A2 A3
AS1
AS2
AS3
AS4
AS5

24. Independent II Begin Algebra;
A= X*X' + T*T'; ! additive genetic covariance
C= Y*Y' + U*U'; ! shared environment covariance
E= Z*Z' + V*V'; ! specific environment covariance
End Algebra;
R=A+C+E; ! total variance
S=(\sqrt(I.R))~; ! diagonal matrix of standard deviations
P=S*X| S*Y| S*Z| \d2v(S*T)'| \d2v(S*U)'| \d2v(S*V)';
! standardized estimates for common and specific factors
\d2v : diagonal to vector

25. Path Diagram to Matrices
Variance Component a2 c2 e2
Common Factors
[X]full
5 x 1
[Y]full
[Z]full
Residual Factors
[T]diag
5 x 5
[U]diag
[V]diag
#define nvar 5
#define nfac 1

26. Path Diagram to Matrices
Variance Component a2 c2 e2
Common Factors
[X]full
5 x 3
[Y]
[Z]full
5 x 1
Residual Factors
[T]diag
5 x 5
[U]
[V]diag
#define nvar 5
#define nfac 1

27. Phenotypic Single Factor
1.00

28. Latent Phenotype T1 P2 A P1 P3 P4 T2 C F1
1.00
1.0 / 0.5 E

29. Factor on Latent Phenotype

30. Common Pathway Model T1 P2 C A E P1 P3 P4 T2 A1 A2 A3 A4 E1 E2 E3 E4 L
1.00
1.00 / 0.50
[T]
[V]
constraint
[X]
[Y]
[Z]
[F]
[U]

31. Common Pathway Example
Script: f:\hmaes\a17\comp.mx
3 MRI measures – 2 IQ subtests:
var1 = cerebellum
var2 = grey matter
var3 = white matter
var4 = calculation
var5 = letters and numbers
Data: f:\hmaes\a17\mri-iqfactor.rec
T:\hmaes/a17\mri-iq-mz.dat
T:\hmaes\a17\mri-iq-dz.dat

32. Compath Pathway #define nvar 5 #define nfac 1 Begin Matrices;
X full nfac nfac Free ! latent factor genetic paths
Y full nfac nfac ! latent factor shared env paths
Z full nfac nfac Free ! latent factor specific env paths
T diag nvar nvar Free ! variable specific genetic paths
U diag nvar nvar ! variable specific shared env paths
V diag nvar nvar Free ! variable specific residual paths
F full nvar nfac Free ! loadings on latent factor
M full 1 nvar Free ! grand means
End Matrices;

33. Compath II Begin Algebra;
A= F&(X*X') + T*T'; ! genetic variance components
C= F&(Y*Y') + U*U'; ! shared environment covariance
E= F&(Z*Z') + V*V'; ! specific environment covariance
L= X*X' + Y*Y' + Z*Z'; ! variance of latent factor
End Algebra;
R=A+C+E; ! total variance
S=(\sqrt(D.R))~; ! diagonal matrix of standard deviations
P=S*F| \d2v(S*T)'| \d2v(S*U)'| \d2v(S*V)';
! standardized estimates for loadings and specific factors
W= X*X' | Y*Y'| Z*Z';
! standardized estimates for latent phenotype

34. Path Diagram to Matrices
Variance Component a2 c2 e2
Common Factor
[X]full
1 x 1
[Y]full
[Z]full
[F]full
5 x 1
Residual Factors
[T]diag
5 x 5
[U]diag
[V]diag
#define nvar 5
#define nfac 1

35. Path Diagram to Matrices
Variance Component a2 c2 e2
Common Factor
[X]full
1 x 1
[Y]
[Z]full
[F]full
5 x 1
Residual Factors
[T]diag
5 x 5
[U]
[V]diag
#define nvar 5
#define nfac 1

36. Summary Independent Pathway Model Common Pathway Model
Biometric Factors Model
Loadings differ for genetic and environmental common factors
Common Pathway Model
Psychometric Factors Model
Loadings equal for genetic and environmental common factor