Multivariate Genetic Analysis (Introduction)

Multivariate Genetic Analysis

(Introduction)

Frühling Rijsdijk

Wednesday March 8, 2006

Multivariate Twin Analyses

Goal: to understand what factors make sets of variables correlate or co-varyTwo or more traits can be correlated because they share common genetic or common environmental influences (C or E)With twin data on multiple traits it’s possible to partition the covariation into it’s genetic and environmental components

Example 1

ADHD IQ

A1 C1 A2 C2

E2E1

rg rc

re

e11 e22

a11 c11 a22 c22

How can we explain the association?

- Additive Genetic effects (rg)

- Shared environment (rc)

- Non-shared environment (re)

Kuntsi et al. Neuropsychiatric Genetics,

Interested in reason for covariance / correlation between phenotypes, e.g. IQ and ADHD

Observed Cov matrices: 4×4Twin1

p1 p2

Within-Twin Covariances

Var P1

Cov P1- P2 Var P2

Twin2 p1

p2

Within-Twin CovariancesCross-Twin Covariances

Within P1

Cross Traits Within P2

Twin1 p1

p2

Twin2

p1 p2

Cov P1- P2 Var P2

Var P1Cross Traits

Twin 1Phenotype 1

Twin 1Phenotype 2

A1 A2C1 C2

E1 E2

a11a22a21

c11 c22

c21

e11 e21e22

Twin 1Phenotype 1

Twin 1Phenotype 2

A1 A2C1 C2

E1 E2

a11a22a21

c11 c22

c21

e11 e21e22

Twin 2Phenotype 1

Twin 2Phenotype 2

A1 A2C1 C2

E1 E2

a11a22a21

c11 c22

c21

e11 e21e22

mz=1; dz=.5 mz=1; dz=.5

1 1

Cholesky Decomposition: Path Tracing


Twin

1

P2

Twin1

p1 p2

A1 A2

a11

P11 P21

a22a21

P1 a112

a222 + a21

2

a11a21


Twin

1

P2

Twin1

p1 p2

P1

C1 C2

c11

P11 P21

c22c21

a112

a222 + a21

2

a11a21

+ c112

+ c222 + c21

2+ c11c21


Twin

1

P2

a112

a222 + a21

2

Twin1

p1 p2

a11a21

P1

P11 P21

E1 E2e11 e21 e22

+ c112 + e11

2

+ e11e21+ c11c21+ e22

2 + e212

+ c222 + c21

2

Cross-Twin Covariances

Twin

2

P2

1/.5a112

1/.5a222 + 1/.5a21

2

Twin1

p1 p2

1/.5a11a21

P1

1/.5 1/.5

A1 A2

a11

P11 P21

a22a21

A1 A2

a11

P12 P22

a22a21

Cross-Twin Covariances

Twin

2

P2

1/.5a112

1/.5a222 + 1/.5a21

2

Twin1

p1 p2

1/.5a11a21

P1 + c112

+ c11c21 + c222 + c21

2

1 1

C1 C2

c11

P11 P21

c22c21

C1 C2

c11

P12 P22

c22c21

Predicted ModelTwin1

p1 p2


Var P1

Cov P1- P2 Var P2

Twin2 p1

p2


Within P1


Twin1 p1

p2

Twin2

p1 p2

Cov P1- P2 Var P2

Var P1

Var of P1 and P2 same across twins and zygosity groups


p1 p2


Var P1

Cov P1- P2 Var P2

Twin2 p1

p2


Within Trait 1

Cross Traits Within Trait 2

Twin1 p1

p2

Twin2

p1 p2

Cov P1- P2 Var P2

Var P1

Cov P1 - P2 same across twins and zygosity groups


p1 p2


Var P1

Cov P1- P2 Var P2

Twin2 p1

p2


Within P1


Twin1 p1

p2

Twin2

p1 p2

Cov P1- P2 Var P2

Var P1

Cross Twin Cov within each trait, different for MZ and DZ


p1 p2


Var P1

Cov P1- P2 Var P2

Twin2 p1

p2


Within P1


Twin1 p1

p2

Twin2

p1 p2

Cov P1- P2 Var P2

Var P1

Cross Twin - Cross trait, different for MZ and DZ


.30

1

1

Twin1p1 p2

Twin2p1 p2

Twin

1

p2

p1Tw

in 2

p2

p1

.59

. 79

.50

.49

. 29

1

1

MZ



.30

1

1

Twin1p1 p2

Twin2p1 p2

Twin

1

p2

p1

Twin

2

p2

p1

.43

.39

.24

.25

. 31

1

1

DZ



.30

1

1

Twin1p1 p2

Twin2p1 p2

Twin

1

p2

p1Tw

in 2

p2

p1

.59

. 79

.25

.24

. 29

1

1

MZ



.30

1

1

Twin1p1 p2

Twin2p1 p2

Twin

1

p2

p1

Twin

2

p2

p1

.43

.39

.24

.23

. 31

1

1

DZ



.30

1

1

Twin1p1 p2

Twin2p1 p2

Twin

1

p2

p1Tw

in 2

p2

p1

.59

. 79

.01

.01

. 29

1

1

MZ



.30

1

1

Twin1p1 p2

Twin2p1 p2

Twin

1

p2

p1

Twin

2

p2

p1

.43

.39

.01

.01

. 31

1

1

DZ


Summary

• Within-individual cross-trait covariance implies common etiological influences

• Cross-twin cross-trait covariance implies that these common etiological influences are familial

• Whether these common familial influences are genetic or environmental, is reflected in the MZ/DZ ratio of the cross-twin cross-traits covariances

Cholesky Decomposition: Specification in Mx

Mx: Parameter Matrices#define nvar 2Begin Matrices;X lower nvar nvar free ! Genetic coefficientsY lower nvar nvar free ! C coefficientsZ lower nvar nvar free ! E coefficientsG Full 1 nvar free ! means End Matrices;Begin Algebra;A=X*X’; ! Gen var/covC=Y*Y’; ! C var/covE=Z*Z’; ! E var/covP=A+C+EEnd Algebra;

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+=Σ

22211121

211111

22

2

aaaaaaa

A

Within-Twin Covariances Path Tracing

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+×+

×+×+=⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡==Σ

2221221121

22211111

22

2111

2221

1122

2

aaa0aa

a0aa00aa0aa

aa0a

XXA *'*

* or ‘Star’ Matrix Multiplication

P1 P2

a22a11

A1 A2

a21

X LOWER 2 × 2 P1P2

A1 A2a11 0a21 a22

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+==Σ

22211121

21111122

2

aaaaaaaXXA '*

Σ P = Σ A + Σ C + Σ E

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+==Σ

22211121

211111

22

2

eeeeeee

ZZE '*

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+++ ++++++++

=Σ22

221

222

221

222

221

2

222

eeccaaeeccaaeeccaaeca

P211111211121

211121112111111111

By rule of matrix addition:

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+==Σ

22211121

211111

22

2

ccccccc

YYC '*

Within-Traits (diagonals):

P11-P12= .5 a112

P21-P22= .5 a222 + .5 a21

2

Cross-Traits:

P11-P22 = .5 a11 a21

P21-P12 = .5 a21 a11

Cross-Twins Covariances, Genetic effects (DZ)

.5 .5

A1 A2

a11

P11 P21

a22a21

A1 A2

a11

P12 P22

a22a21

Twin 1 Twin 2

( )⎥⎥⎦⎤

⎢⎢⎣

⎡

+=⊗=Σ⊗

222122

2

aa5aa5aa5a5XX5A5

1121

211111

....'*..

Path Tracing

Kronecker Product ⊗ @

Cross-Twins Covariances, Genetic effects (MZ)

1 1

A1 A2

a11

P11 P21

a22a21

A1 A2

a11

P12 P22

a22a21

Twin 1 Twin 2

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+=⊗=Σ⊗

22211121

11

22

2

aaaaaaaXX1A1 2111'*

Cross-Twins Covariances, C effects (MZ and DZ)

1 1

C1 C2

c11

P11 P21

c22c21

C1 C2

c11

P12 P22

c22c21

Twin 1 Twin 2

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+=⊗=Σ⊗

22211121

11

22

2

cccccccYY1C1 2111'*

Covariance Model for Twin pairs

Covariance A+C+E | A+C _A+C | A+C+E /

Covariance A+C+E | H@A+C _H@A+C | A+C+E /

MZ

DZ

Standardized Estimates

Correlated Factors Solution

P1t1 P2t1

A1 C1 A2 C2

E2E1

rg rc

re

e11 e22

a11 c11 a22 c22

Covariances to Correlations

In matrix form:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

σ

σ⎥⎦

⎤⎢⎣

⎡σσσσ

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

σ

σ=⎥⎦

⎤⎢⎣

⎡

222

112

222212

122112

222

112

21

1210

01

10

01

1rr1

**

222

112

122

12r

σσ

σ=

* 222212

1121211

rσ

σσ

= **

Genetic Correlations

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+=Σ

222

2211121

2111211

aaaa

aaaA

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

σ

σ⎥⎦

⎤⎢⎣

⎡

σσσσ

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

σ

σ=⎥⎦

⎤⎢⎣

⎡

22A2

11A2

22A2

21A2

12A2

11A2

22A2

11A2

10

01

10

01

1rr1

G

G **

Specification in Mx

Matrix Function in Mx: R = \stnd (A);

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

σ

σ⎥⎦

⎤⎢⎣

⎡

σσσσ

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

σ

σ=⎥⎦

⎤⎢⎣

⎡

22A2

11A2

22A2

21A2

12A2

11A2

22A2

11A2

10

01

10

01

1rr1

G

G **

R = \sqrt( I . A )˜ * A * \sqrt( I . A )˜;

Where I is an Identity matrix :

1

2

1 00 1

σ2A11 0

0 σ2A22

And I.A =

InterpretationHigh Genetic correlation: large overlap in genetic effects on the two traits

Does it mean that the phenotypic correlation between the traits is largely due to genetic effects?

No, the substantive importance of a particular rgdepends both on the value of the correlation and the value of the A paths, i.e. the heritabilities of both traits.

A1 A2

√h2

P1 P2

√h2

rg

(√h2p1 * rg * √h2

p2) / rPH

(√.63 * -0.525 * √.33) / -0.29 = .8357

Interpretation• For Example, consider a phenotypic correlation of 0.40

With √h2p1 = .7 and √h2

p2 = .6 with rg = .3.19 (49%) of the phenotypic correlation can be attributed toadditive genetic effects.

With √h2p1 = .2 and √h2

p2 = .4 with rg = .8.20 (49%) of the phenotypic correlation can be attributed toadditive genetic effects.

• Weakly heritable traits can still have a large portion of their correlation attributable to genetic effects.

More variables……

Twin 1Phenotype 1

Twin 1Phenotype 2

A1 A2C1 C2

E1 E2

a11

a22a21c11

c22c21

e11

e21e22

Twin 1Phenotype 3

A3 C3

a33

E3

e33

c33

c31a23 c23

a31

e31e23

More variables……

Twin 1Phenotype 1

Twin 1Phenotype 2

A1 A2C1 C2

E1 E2

a11

a22a21c11

c22c21

e11

e21e22

Twin 1Phenotype 3

A3 C3

a33

E3

e33

c33

c31a23 c23

a31

e31e23

e22

Twin 2Phenotype 1

Twin 2Phenotype 2

A1 A2C1 C2

E1 E2

a11

a22a21c11

c22c21

e11

e21e22

Twin 2Phenotype 3

A3 C3

a33

E3

e33

c33

c31a23 c23

a31

e31e23

e22

Mx: Parameter Matrices#define nvar 3Begin Matrices;X lower nvar nvar free ! Genetic coefficientsY lower nvar nvar free ! C coefficientsZ lower nvar nvar free ! E coefficientsG Full 1 nvar free ! means End Matrices;Begin Algebra;A=X*X’; ! Gen var/covC=Y*Y’; ! C var/covE=Z*Z’; ! E var/covP=A+C+EEnd Algebra;

X LOWER 3 × 3

A1 A2 A3a11 0 0a21 a22 0a31 a32 a33

Y LOWER 3 × 3

C1 C2 C3c11 0 0c21 c22 0c31 c32 c33

Z LOWER 3 × 3

E1 E2 E3e11 0 0e21 e22 0e31 e32 e33

Multivariate Genetic Analysis (Introduction)

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