Multivariate Analysis

Multivariate AnalysisHGEN619 class 2007

HGEN61910/20/03

Multivariate Questions I Bivariate Analysis: What are the contributions of

genetic and environmental factors to the covariance between two traits?

Multivariate Analysis: What are the contributions of genetic and environmental factors to the covariance between more than two traits?

Phenotypic Cholesky F1

F1 F4F3F2P1

P4P3P2

f11

f41f31f21

F1 F2

f11

P1t1

1 1

P2t1

f21

F31

P3t1

F41

P4t1

f31 f41

Phenotypic Cholesky F2

F1 F4F3F2P1

P4P3P2

f11

f41f31f21 f22

f42f32

0F1 F2

f22

P1t1

1 1

P2t1

F31

P3t1

F41

P4t1

f32 f42

Phenotypic Cholesky

F1 F4F3F2P1

P4P3P2

f11

f41f31f21 f22

f33f42f32

f43

000

F1 F2

P1t1

1 1

P2t1

F3

f33

1

P3t1

F41

P4t1

f43

Phenotypic Cholesky

F1 F4F3F2P1

P4P3P2

f11

f41f31f21 f22

f33f42f32

f43 f44

0

000

0 0F1 F2

P1t1

1 1

P2t1

F31

P3t1

F4

f44

1

P4t1

Cholesky Decomposition

f11 f41f31f21f22

f33f42f32f43f44

0

0000

0

F1 F4F3F2P1

P4P3P2

f11

f41f31f21 f22

f33f42f32

f43 f44

0

000

0 0

*F F'

*

Saturated Model

Use Cholesky decomposition to estimate covariance matrix

Fully saturated Model: Cov P = F*F’

F: Full nvar nvar

Phenotypic Single Factor

F1P1

P4P3P2

f11

f41f31f21 *

f11 f21 f31 f41

F * F'

F1

f11

P1t1

1

P2t1

f21

P3t1 P4t1

f31 f41

Residual VariancesE1 E4E3E2

P1

P4P3P2

e11e22

e33e44

0

000

0 0

00

000

0*

e11e22

e33e44

0

000

0 0

00

000

0

*E E'f11

P1t1 P2t1

f21

P3t1 P4t1

f31 f41

F11

E11

E41

E31

E21

e11 e44e33e22

Factor Analysis

Explain covariance by limited number of factors

Exploratory / Confirmatory Model: Cov P = F*F’ + E*E’

F: Full nvar nfacE: Diag nvar nvar

Twin Data

f11

P1t1 P2t1

f21

P3t1 P4t1

f31 f41

F11

E11

E41

E31

E21

e11 e44e33e22

f11

P1t2 P2t2

f21

P3t2 P4t2

f31 f41

F11

E11

E41

E31

E21

e11 e44e33e22

?

Genetic Single Factor

a11

P1t1 P2t1

a21

P3t1 P4t1

a31 a41

A11

E11

E41

E31

E21

e11 e44e33e22

a11

P1t2 P2t2

a21

P3t2 P4t2

a31 a41

A11

E11

E41

E31

E21

e11 e44e33e22

1.0 / 0.5

Single [Common] Factor X: genetic

Full 4 x 1 Full nvar x nfac

Y: shared environmental Z: specific environmental

A1P1

P4P3P2

a11

a41a31a21 *

a11a21a31a41

X * X'

Common Environmental Single Factor

P1t1 P2t1 P3t1 P4t1

A11

E11

E41

E31

E21

e11 e44e33e22

P1t2 P2t2 P3t2 P4t2

A11

E11

E41

E31

E21

e11 e44e33e22

1.0 / 0.5

c11 c21 c31 c41

C11

c11 c21 c31 c41

C11

1.0

Specific Environmental Single Factor

P1t1 P2t1 P3t1 P4t1

A11

E11

E41

E31

E21

e11 e44e33e22

P1t2 P2t2 P3t2 P4t2

A11

E11

E41

E31

E21

e11 e44e33e22

1.0 / 0.5

C11

C11

1.0

e11 e21 e31e41

E11

e11 e21 e31e41

E11

Residuals partitioned in ACE

P1t1 P2t1 P3t1 P4t1

A11

E11

E41

E31

E21

e11 e44e33e22

P1t2 P2t2 P3t2 P4t2

A11

E11

E41

E31

E21

e11 e44e33e22

1.0 / 0.5

C11

C11

1.0

E11

E11

C11

c11

A11

a11

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

Diag 4 x 4 Diag nvar x nvar E1 E4E3E2

P1

P4P3P2

e11e22

e33e44

0

000

0 0

00

000

0*

e11e22

e33e44

0

000

0 0

00

000

0

*V V'

Independent Pathway Model

P1t1 P2t1 P3t1 P4t1

AC

1

P1t2 P2t2 P3t2 P4t2

AC

1

1.0 / 0.5

CC

1

CC

1

1.0

EC

1

EC

1

C11

E1

1

A11

E2

1

A21

E3

1

A31

E4

1

A41

E1

1

A11

E2

1

A21

E3

1

A31

E4

1

A41

1.0 / 0.5 1.0 / 0.5 1.0 / 0.5 1.0 / 0.5

[X] [Y] [Z]

[T]

[V][U]

Path Diagram to MatricesVariance Component

a2 c2 e2

Common Factors

[X]6 x 1

[Y]6 x 1

[Z]6 x 1

Residual Factors

[T]6 x 6

[U]6 x 6

[V]6 x 6

#define nvar 6#define nfac 1

Independent Pathway I G1: Define matrices Calculation Begin Matrices; X full nvar nfac Free ! common factor genetic path coefficients Y full nvar nfac Free ! common factor shared environment paths Z full nvar nfac Free ! common factor unique environment paths T diag nvar nvar Free ! variable specific genetic paths U diag nvar nvar Free ! variable specific shared env paths V diag nvar nvar Free ! variable specific residual paths M full 1 nvar Free ! means End Matrices; Start … Begin Algebra; A= X*X' + T*T'; ! additive genetic variance components C= Y*Y' + U*U'; ! shared environment variance components E= Z*Z' + V*V'; ! nonshared environment variance components End Algebra; End

indpath.mx

Independent Pathway II G2: MZ twins #include iqnlmz.dat Begin Matrices = Group 1; Means M | M ; Covariance A+C+E | A+C _ A+C | A+C+E ; Option Rsiduals End

G3: DZ twins #include iqnldz.dat Begin Matrices= Group 1; H full 1 1 End Matrices; Matrix H .5 Means M | M ; Covariance A+C+E | H@A+C _ H@A+C | A+C+E ; Option Rsiduals End

Independent Pathway III G4: Calculate Standardised Solution Calculation Matrices = Group 1 I Iden nvar nvar End Matrices; 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 for common factors Q=S*T_ S*U_ S*V; ! standardized estimates for spec factors End Algebra; Labels Row P a1 a2 a3 a4 a5 a6 c1 c2 c3 c4 c5 c6 e1 e2 e3 e4 e5 e6 Labels Col P var1 var2 var3 var4 var5 var6 Labels Row Q as1 as2 as3 as4 as5 as6 cs1 cs2 cs3 cs4 cs5 cs6 es1 es2

es3 es4 es5 es6 Labels Col Q var1 var2 var3 var4 var5 var6 Options NDecimals=4 End

IP

Independent pathwaysBiometric modelDifferent covariance structure for A, C and E

Phenotypic Single Factor

F1P1

P4P3P2

f11

f41f31f21 *

f11 f21 f31 f41

F * F'

F1

f11

P1t1

1

P2t1

f21

P3t1 P4t1

f31 f41

Latent Phenotype

F1

f11

P1t1 P2t1

f21

P3t1 P4t1

f31 f41

E1

C1

A1

eca

Twin Data

F1

f11

P1t1 P2t1

f21

P3t1 P4t1

f31 f41

E1

C1

A1

eca

F1

f11

P1t1 P2t1

f21

P3t1 P4t1

f31 f41

E1

C1

A1

eca

1.0 / 0.5 1.0

Factor on Latent PhenotypeF1

P1

P4P3P2

f11

f41f31f21 * f11 f21 f31 f41

F * F'

a a* *

X X'* *

F & X X'*( )=

Common Pathway Model

F1

f11

P1t1 P2t1

f21

P3t1 P4t1

f31 f41

E1

C1

A1

eca

F1

f11

P1t1 P2t1

f21

P3t1 P4t1

f31 f41

E1

C1

A1

eca

1.0 / 0.5 1.0

C11

E1

1

A11

E2

1

A21

E3

1

A31

E4

1

A41

E1

1

A11

E2

1

A21

E3

1

A31

E4

1

A41

1.0 / 0.5 1.0 / 0.5 1.0 / 0.5 1.0 / 0.5

[T]

[V][U]

[X] [Y] [Z]

[F]

Path Diagram to MatricesVariance Component

a2 c2 e2

Common Factor

[X]1 x 1

[Y]1 x 1

[Z]1 x 1

[F]6 x 1

Residual Factors

[T]6 x 6

[U]6 x 6

[V]6 x 6

#define nvar 6#define nfac 1

Common Pathway Model I G1: Define matrices Calculation Begin Matrices; X full nfac nfac Free ! latent factor genetic path coefficient Y full nfac nfac Free ! latent factor shared environment path Z full nfac nfac Free ! latent factor unique environment path T diag nvar nvar Free ! variable specific genetic paths U diag nvar nvar Free ! variable specific shared env paths V diag nvar nvar Free ! variable specific residual paths F full nvar nfac Free ! loadings of variables on latent factor I Iden 2 2 M full 1 nvar Free ! means End Matrices; Start .. Begin Algebra; A= F&(X*X') + T*T'; ! genetic variance components C= F&(Y*Y') + U*U'; ! shared environment variance components E= F&(Z*Z') + V*V'; ! nonshared environment variance components L= X*X' + Y*Y' + Z*Z'; ! variance of latent factor End Algebra; End

compath.mx

Common Pathway II G4: Constrain variance of latent factor to 1 Constraint Begin Matrices; L computed =L1 I unit 1 1 End Matrices; Constraint L = I ; End

G5: Calculate Standardised Solution Calculation Matrices = Group 1 D Iden nvar nvar End Matrices; Begin Algebra; R=A+C+E; ! total variance S=(\sqrt(D.R))~; ! diagonal matrix of standard deviations P=S*F; ! standardized estimates for loadings on F Q=S*T_ S*U_ S*V; ! standardized estimates for specific factors End Algebra; Options NDecimals=4 End

CP

Common pathwayPsychometric modelSame covariance structure for A, C and E

Practical Example Dataset: NL-IQ Study 6 WAIS-III IQ subtests

var1 = onvolledige tekeningen / picture completion var2 = woordenschat / vocabulary var3 = paren associeren / digit span var4 = incidenteel leren / incidental learning var5 = overeenkomsten / similarities var6 = blokpatronen / block design

N MZF: 27, DZF: 70

Independent Pathway ModelBiometric Factor ModelLoadings differ for genetic and environmental

common factors Common Pathway Model

Psychometric Factor ModelLoadings equal for genetic and environmental

common factor

Summary

WAIS-III IQ

Verbal IQ var2 = woordenschat / vocabulary var3 = paren associeren / digit span var5 = overeenkomsten / similarities

Performance IQ var1 = onvolledige tekeningen / picture completion var4 = incidenteel leren / incidental learning var6 = blokpatronen / block design

Pathway Model

F2

P1t1 P2t1 P3t1 P4t1

a21

f11

P1t2 P2t2

f21

P3t2 P4t2

f31 f41

1.0 / 0.5

E1

A1

E2 E3 E4 E1

1

A11

E51

E61

1.0 / 0.5[T]

[Z]

[F]

P5t2 P6t2P6t1P5t1

F1

f11 f21 f31 f41 f51 f1

F3

f61

E11

C11

A11

E21

C21

A21

F1

E1

C1

A1

eca

E31

C31

A31

a32a31 a33a11 a22

1

f51 f61

1.0

1

111C1

1C1

1.0 1[U]

[Y][X]

F2 F3

[V]

Two Common Pathway Model

F2

P1t1 P2t1 P3t1 P4t1

a21

f11

P1t2 P2t2

f21

P3t2 P4t2

f31 f41

1.0 / 0.5

E1

A1

E2 E3 E4 E1

1

A11

E51

E61

1.0 / 0.5[T]

[Z]

[F]

P5t2 P6t2P6t1P5t1

F1

f11 f21 f31 f41 f51 f1

F3

E11

C11

A11

E21

C21

A21

F1

E1

C1

A1

eca

E31

C31

A31

1a11 a22

1

f51 f61

1.0

1

111C1

1C1

1.0 1[U]

[Y][X]

F2 F3

[V]

Two Independent CP Model

F2

P1t1 P2t1 P3t1 P4t1

f11

P1t2 P2t2

f21

P3t2 P4t2

f31 f41

1.0 / 0.5

E1

A1

E2 E3 E4 E1

1

A11

E51

E61

1.0 / 0.5[T]

[Z]

[F]

P5t2 P6t2P6t1P5t1

F1

f11 f21 f31 f41 f51 f1

F3

E11

C11

A11

E21

C21

A21

F1

E1

C1

A1

eca

E31

C31

A31

1a11 a22

1

f51 f61

1.0

1

111C1

1C1

1.0 1[U]

[Y][X]

F2 F3

[V]

Two Reduced Indep CP Model

F2

f12

P1t1 P2t1 P3t1 P4t1

f42 f11

P1t2 P2t2

f21

P3t2 P4t2

f31 f41

1.0 / 0.5

E1

A1

E2 E3 E4 E1

1

A11

E51

E61

1.0 / 0.5[T]

[Z]

[F]

P5t2 P6t2P6t1P5t1

f62

F1

f21 f31 f51

F3

E11

C11

A11

E21

C21

A21

F1

E1

C1

A1

eca

E311

A31

1a11 a22

1

f51 f61

1.0

1

111C1

1C1

1.0 1[U]

[Y][X]

F2 F3

[V]

C3

Common Pathway Model

1

F2

f11

P1t1 P2t1 P3t1 P4t1

f41 f11

P1t2 P2t2

f21

P3t2 P4t2

f31 f41

1.0 / 0.5

E1

A1

E2 E3 E4 E1

1

A11

E51

E61

1.0 / 0.5[T]

[Z]

[F]

P5t2 P6t2P6t1P5t1

f61

F1

f21 f31 f51

F3

E11

C11

A11

E21

C21

A21

F1

E1

C1

A1

eca

E311

A31

a11

1

f51 f61

1.0

1

111C1

1C1

1.0 1[U]

[Y][X]

F2 F3

[V]

C3

1

Independent Pathway Model

F2

P1t1 P2t1 P3t1 P4t1

f11

P1t2 P2t2

f21

P3t2 P4t2

f31 f41

1.0 / 0.5

E1

A1

E2 E3 E4 E1

1

A11

E51

E61

1.0 / 0.5[T]

[Z]

[F]

P5t2 P6t2P6t1P5t1

F1

f11 f21 f31 f41 f51 f1

F3

f61

E11

C11

A11

E21

C21

A21

F1

E1

C1

A1

111

E31

C31

A31

1 11

1

f51 f61

1.0

1

111C1

1C1

1.0 1[U]

[Y][X]

F2 F3

[V]