Multivariate Genetic Analysis

Multivariate Genetic Analysis

Boulder 2004

Phenotypic Cholesky

T1P2

F1 F4

T1P1 T1P3 T1P4

F2 F3

1.00 1.001.00 1.00

F1 F4F3F2

P1

P4

P3

P2

f11

f41

f31

f21

Phenotypic Cholesky

T1P2

F1 F4

T1P1 T1P3 T1P4

F2 F3

1.00 1.001.00 1.00

F1 F4F3F2

P1

P4

P3

P2

f11

f41

f31

f21 f22

f42

f32

0

Phenotypic Cholesky

T1P2

F1 F4

T1P1 T1P3 T1P4

F2 F3

1.00 1.001.00 1.00

F1 F4F3F2

P1

P4

P3

P2

f11

f41

f31

f21 f22

f33

f42

f32

f43

0

0

0

Phenotypic Cholesky

T1P2

F1 F4

T1P1 T1P3 T1P4

F2 F3

1.00 1.001.00 1.00

F1 F4F3F2

P1

P4

P3

P2

f11

f41

f31

f21 f22

f33

f42

f32

f43 f44

0

0

00

0 0

Cholesky Decomposition

f11 f41f31f21

f22

f33

f42f32

f43

f44

0

00

00

0

F1 F4F3F2

P1

P4

P3

P2

f11

f41

f31

f21 f22

f33

f42

f32

f43 f44

0

0

00

0 0

*F F'

*

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

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;

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

Phenotypic Single Factor

T1P2

F1

T1P1 T1P3 T1P4

1.00F1

P1

P4

P3

P2

f11

f41

f31

f21*

f11 f21 f31 f41

F * F'

Residual Variances

T1P2

F1

T1P1 T1P3 T1P4

1.00

1.00

E1

1.00

E2

1.00

E3

1.00

E4

E1 E4E3E2

P1

P4

P3

P2

e11

e22

e33

e44

0

0

00

0 0

0

0

0

0

0

0

*

e11

e22

e33

e44

0

0

00

0 0

0

0

0

0

0

0

*E E'

Twin Data

T1P2

F1 F1

T1P1 T1P3 T1P4 T2P2T2P1 T2P3 T2P4

E1 E2 E3 E4 E1 E2 E3 E4

1.00 1.00?

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

Genetic Single Factor

T1P2

A1 A1

T1P1 T1P3 T1P4 T2P2T2P1 T2P3 T2P4

E1 E2 E3 E4 E1 E2 E3 E4

1.00 1.00

1.0 / 0.5

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

Single [Common] Factor X: genetic

Full 4 x 1 Full nvar x nfac

Y: shared environmental Z: specific environmental

A1

P1

P4

P3

P2

a11

a41

a31

a21*

a11 a21 a31 a41

X * X'

Common Environmental Single Factor

T1P2

A1 A1

T1P1 T1P3 T1P4 T2P2T2P1 T2P3 T2P4

E1 E2 E3 E4 E1 E2 E3 E4

1.00 1.00

1.0 / 0.5

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.00

C1

1.00

C1

1.00

Specific Environmental Single Factor

T1P2

A1 A1

T1P1 T1P3 T1P4 T2P2T2P1 T2P3 T2P4

E1 E2 E3 E4 E1 E2 E3 E4

1.00 1.00

1.0 / 0.5

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.00

C1

1.00

C1

1.00

1.00

E1 E1

1.00

Residuals partitioned in ACE

T1P2

A1 A1

T1P1 T1P3 T1P4 T2P2T2P1 T2P3 T2P4

E2 E3 E4 E1 E2 E3 E4

C1 C1E1 E1

1.00 1.00

1.0 / 0.5

1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.00 1.00

1.00

1.00 1.00

E1

1.00

A1

1.00C1

1.00

Residual Factors T: genetic U: shared environmental V: specific environmental

Diag 4 x 4 Diag nvar x nvar E1 E4E3E2

P1

P4

P3

P2

e11

e22

e33

e44

0

0

00

0 0

0

0

0

0

0

0

*

e11

e22

e33

e44

0

0

00

0 0

0

0

0

0

0

0

*V V'

Independent Pathway Model

T1P2

CA CC CE CA CC CE

T1P1 T1P3 T1P4 T2P2T2P1 T2P3 T2P4

A1 A2 A3 A4 A1 A2 A3 A4

E1 E2 E3 E4 E1 E2 E3 E4C1

1.00 1.00 1.00 1.00 1.00 1.00

1.00 / 0.50 1.00

[X][Y]

[Z]

1.00 1.00 1.00 1.00

[T]

1.00 1.00 1.00 1.00

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

[V]

1.00 / 0.501.00 / 0.501.00 / 0.501.00 / 0.50

1.00

[U]

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

Independent Pathway#define nvar 5#define nfac 1

G1: Define matrices Calculation Begin Matrices; X full nvar 3 ! 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;

Three Genetic Factors

Specify X

! declare free parameters for 3 genetic common factors! G1 G2 G3 100 110 0 101 111 0 102 112 0 103 0 120 104 0 121 Drop T 1 5 5 ! fix genetic specific for last variable for identification

Three Genetic Factors

CB GM WM RK CL

1.00

A11.00

A2

1.00

A3

1.00

AS1

1.00

AS2

1.00

AS3

1.00

AS4

1.00

AS5

Independent IIBegin Algebra; A= X*X' + T*T'; ! additive genetic covariance C= Y*Y' + U*U'; ! shared environment covariance E= Z*Z' + V*V'; ! specific environment covarianceEnd Algebra;

Begin 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 End Algebra;

\d2v : diagonal to vector

Path Diagram to Matrices

Variance Component

a2 c2 e2

Common Factors

[X]full5 x 1

[Y]full5 x 1

[Z]full5 x 1

Residual Factors

[T]diag5 x 5

[U]diag5 x 5

[V]diag5 x 5

#define nvar 5#define nfac 1

Path Diagram to Matrices

Variance Component

a2 c2 e2

Common Factors

[X]full5 x 3

[Y] [Z]full5 x 1

Residual Factors

[T]diag5 x 5

[U] [V]diag5 x 5

#define nvar 5#define nfac 1

Phenotypic Single Factor

T1P2

F1

T1P1 T1P3 T1P4

1.00F1

P1

P4

P3

P2

f11

f41

f31

f21*

f11 f21 f31 f41

F * F'

Latent Phenotype

T1P2

A A

T1P1 T1P3 T1P4 T2P2T2P1 T2P3 T2P4

C

F1 F1

1.00 1.00

1.0 / 0.5

1.00

E

1.00

C E

1.00 1.00

1.00

Factor on Latent Phenotype

F1

P1

P4

P3

P2

f11

f41

f31

f21* f11 f21 f31 f41

F * F'

a a* *

X X'* *

F & X X'*( )=

Common Pathway Model

T1P2

CA CC CE CA CC CE

T1P1 T1P3 T1P4 T2P2T2P1 T2P3 T2P4

A1 A2 A3 A4 A1 A2 A3 A4

E1 E2 E3 E4 E1 E2 E3 E4

L L

C1

1.00 1.00 1.00 1.00 1.00 1.00

1.00 / 0.50 1.00

1.00 1.00 1.00 1.00

[T]

1.00 1.00 1.00 1.00

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

[V]

1.00 / 0.501.00 / 0.501.00 / 0.501.00 / 0.50

constraint constraint

[X][Y]

[Z]

[F]

1.00

[U]

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

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;

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;

Begin 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 End Algebra;

Path Diagram to Matrices

Variance Component

a2 c2 e2

Common Factor

[X]full1 x 1

[Y]full1 x 1

[Z]full1 x 1

[F]full5 x 1

Residual Factors

[T]diag5 x 5

[U]diag5 x 5

[V]diag5 x 5

#define nvar 5#define nfac 1

Path Diagram to Matrices

Variance Component

a2 c2 e2

Common Factor

[X]full1 x 1

[Y] [Z]full1 x 1

[F]full5 x 1

Residual Factors

[T]diag5 x 5

[U] [V]diag5 x 5

#define nvar 5#define nfac 1

Summary Independent 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