Post on 19-Oct-2021
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
Within-Twin Covariances
Twin
1
P2
Twin1
p1 p2
A1 A2
a11
P11 P21
a22a21
P1 a112
a222 + a21
2
a11a21
Within-Twin Covariances
Twin
1
P2
Twin1
p1 p2
P1
C1 C2
c11
P11 P21
c22c21
a112
a222 + a21
2
a11a21
+ c112
+ c222 + c21
2+ c11c21
Within-Twin Covariances
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
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 P1
Var of P1 and P2 same across twins and zygosity groups
Predicted ModelTwin1
p1 p2
Within-Twin Covariances
Var P1
Cov P1- P2 Var P2
Twin2 p1
p2
Within-Twin CovariancesCross-Twin Covariances
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
Predicted ModelTwin1
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 P1
Cross Twin Cov within each trait, different for MZ and DZ
Predicted ModelTwin1
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 P1
Cross Twin - Cross trait, different for MZ and DZ
Within-Twin Covariances
.30
1
1
Twin1p1 p2
Twin2p1 p2
Twin
1
p2
p1Tw
in 2
p2
p1
.59
. 79
.50
.49
. 29
1
1
MZ
Within-Twin CovariancesCross-Twin Covariances
Within-Twin Covariances
.30
1
1
Twin1p1 p2
Twin2p1 p2
Twin
1
p2
p1
Twin
2
p2
p1
.43
.39
.24
.25
. 31
1
1
DZ
Within-Twin CovariancesCross-Twin Covariances
Within-Twin Covariances
.30
1
1
Twin1p1 p2
Twin2p1 p2
Twin
1
p2
p1Tw
in 2
p2
p1
.59
. 79
.25
.24
. 29
1
1
MZ
Within-Twin CovariancesCross-Twin Covariances
Within-Twin Covariances
.30
1
1
Twin1p1 p2
Twin2p1 p2
Twin
1
p2
p1
Twin
2
p2
p1
.43
.39
.24
.23
. 31
1
1
DZ
Within-Twin CovariancesCross-Twin Covariances
Within-Twin Covariances
.30
1
1
Twin1p1 p2
Twin2p1 p2
Twin
1
p2
p1Tw
in 2
p2
p1
.59
. 79
.01
.01
. 29
1
1
MZ
Within-Twin CovariancesCross-Twin Covariances
Within-Twin Covariances
.30
1
1
Twin1p1 p2
Twin2p1 p2
Twin
1
p2
p1
Twin
2
p2
p1
.43
.39
.01
.01
. 31
1
1
DZ
Within-Twin CovariancesCross-Twin Covariances
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