Download - Bivariate analysis
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Bivariate analysis
HGEN619 class 2005
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Univariate ACE model
T 2
AEA C
a ac e
T 1
1 111
E
e
1
C
c
1
1 or .5
1
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Expected Covariance Matrices
a2+c 2+e 2 .5a 2+c 2
.5a 2+c 2 a 2+c 2+e 2 DZ =
a2+c 2+e 2 a2+c 2
a2+c 2 a 2+c 2+e 2 MZ = 2 x 2
2 x 2
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Bivariate Questions I
Univariate Analysis: What are the contributions of additive genetic, dominance/shared environmental and unique environmental factors to the variance?
Bivariate Analysis: What are the contributions of genetic and environmental factors to the covariance between two traits?
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Two Traits
E YE X
X Y
A X A YA C
E C
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Bivariate Questions II
Two or more traits can be correlated because they share common genes or common environmental influences e.g. Are the same genetic/environmental factors
influencing the traits? With twin data on multiple traits it is possible to
partition the covariation into its genetic and environmental components
Goal: to understand what factors make sets of variables correlate or co-vary
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Bivariate Twin Data
individual twin
within between
trait within
between
(within-twin within-trait
co)variance(cross-twin within-trait)
covariance(cross-twin within-trait) covariance
cross-twin cross-trait covariance
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Bivariate Twin Covariance Matrix
X1 Y1 X2 Y2
X1
Y1
X2
Y2
VX1 CX1X2
CX2X1 VX2
VY1 CY1Y2
CY2Y1 VY2
CX1Y1
CX2Y2
CY1X1
CY2X2
CX1Y2
CX2Y1
CY1X2
CY2X1
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Genetic Correlation
Y 2
A YA X A Y
a x a ya y
X 1
1 11
A X
a x
1
1 or .5 1 or .5
X 2Y 1
rg rg
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Alternative Representations
A X A Y
a x a y
X 1
1 1
Y 1
rg
A S X A S Y
a sx a sy
X 1
1 1
Y 1
A C
1
a c a c
A 1 A 2
a11 a22
X 1
1 1
Y 1
a21
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Cholesky Decomposition
A 1 A 2
a11 a22
X 1
1 1
Y 1
a21
A 1 A 2
a11 a22
X 2
1 1
Y 2
a21
1 or .5 1 or .5
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More Variables
A 1 A 2
a11 a22
X 1
1 1
X 2
a21
A 3
a33
1
X 3
A 4
a44
1
X 4
a32 a43a31 a42
A 5
1
X 5
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Bivariate AE Model
A 1 A 2
a11 a22
X 1
1 1
Y 1
a21
A 1 A 2
a11 a22
X 2
1 1
Y 2
a21
1 or .5 1 or .5
E 1 E 2
e11 e22
1 1
e21
E 1 E 2
e11 e22
1 1
e21
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MZ Twin Covariance Matrix
X1 Y1 X2 Y2
X1
Y1
X2
Y2
a112
+e112
a222+a21
2+e22
2+e212
a21*a11+e21*e11
a222+a21
2
a112
a21*a11
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DZ Twin Covariance Matrix
X1 Y1 X2 Y2
X1
Y1
X2
Y2
a112
+e112
a222+a21
2+e22
2+e212
a21*a11+e21*e11
.5a222+
.5a212
.5a112
.5a21*a11
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Within-Twin Covariances [Mx]
A 1 A 2
a11 a22
X 1
1 1
Y 1
a21a11
a22
0
a21
A 1 A 2
X 1
Y 1
X Lower 2 2
a112
a222+a 21
2a21 *a 11
a11 *a 21=
A=X*X'
a11
a22
0
a21
a11
a220
a21
* A=
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Within-Twin Covariances
a112
a222+a 21
2a21 *a 11
a11 *a 21 A=
e112
e222+e 21
2e21 *e 11
e11 *e 21 E=
a112+ e 11
2
a222+a 21
2 +e 222+e 21
2
a11 *a 21 + e 11 *e 21 P= A+ E =a21 *a 11 + e 21 *e 11
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Cross-Twin Covariances
a112
a222+a 21
2a21 *a 11
a11 *a 21 A=MZ
.5a 112
.5a 222+.5a 21
2.5a 21 *a 11
.5a 11 *a 21.5@ A=DZ
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Cross-Trait Covariances
Within-twin cross-trait covariances imply common etiological influences
Cross-twin cross-trait covariances imply familial common etiological influences
MZ/DZ ratio of cross-twin cross-trait covariances reflects whether common etiological influences are genetic or environmental
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Univariate Expected Covariances
a2+c 2+e 2 .5a 2+c 2
.5a 2+c 2 a 2+c 2+e 2 DZ =
a2+c 2+e 2 a2+c 2
a2+c 2 a 2+c 2+e 2 MZ = 2 x 2
2 x 2
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Univariate Expected Covariances II
DZ = A+ C+ E .5@ A+ C.5@ A+ C A+ C+ E
A+ C+ E A+ C A+ C A+ C+ E
MZ = 2 x 2
2 x 2
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Bivariate Expected Covariances
DZ = A+ C+ E .5@ A+ C.5@ A+ C A+ C+ E
A+ C+ E A+ C A+ C A+ C+ E
MZ = 4 x 4
4 x 4
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Practical Example I
Dataset: MCV-CVT Study 1983-1993 BMI, skinfolds (bic,tri,calf,sil,ssc) Longitudinal: 11 years N MZF: 107, DZF: 60
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Practical Example II
Dataset: NL MRI Study 1990’s Working Memory, Gray & White Matter
N MZFY: 68, DZF: 21