multivariate genetic analysis
DESCRIPTION
Multivariate Genetic Analysis. Boulder 2004. 1.00. 1.00. 1.00. 1.00. F1. F2. F3. F4. P1. P2. P3. P4. T1. T1. T1. T1. Phenotypic Cholesky. 1.00. 1.00. 1.00. 1.00. F1. F2. F3. F4. P1. P2. P3. P4. T1. T1. T1. T1. Phenotypic Cholesky. 1.00. 1.00. 1.00. 1.00. F1. - PowerPoint PPT PresentationTRANSCRIPT
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